Agreed, with the caveat that any point exactly on a side or corner of a 2D object will necessarily have fewer degrees (since in at least some of the directions from that point, there is no 2D object).
A circle is not a polygon. It's true that the question is kinda vague. I would answer it so that the circle is 360 and the rest of them, polygons that they are, follow the formula: sum of interior angles = (n-2)*180 where n is the number of sides. If I got docked, I'd complain to the instructor that his test questions are vague.
there are so many problems like that now a days. weve all seen the ones on instagram and tiktok that are something like what is 2 / 4(5-3) but it uses the old divide symbol and doesnt say where the divide actually is. so the answer is both of them it just depends on where you put the brackets. theyre awfully worded and constructed. its mainly just to get engagement
@@alykadane7206 Problem is, then you've got teachers marking answers wrong because they know less about the subject matter than the kids they're "teaching." I had (mostly) fantastic math teachers back when I was in school, but it definitely seems like quality has fallen off a cliff in the past couple decades.
Infinity was the correct answer. 360° was wrong. Unless you think someone has just proved that a circle is a four sided shape. Or, if you interpret the question differently and it's about the number of degrees around the centre, then all 4 shapes have 360°. However you interpret the question you have to apply it to all the shapes the same manner. It is either about internal angles or it is about the shape being closed. This is just another example of the person setting the question having insufficient knowledge about the subject. It's the kind of low quality teaching and wooly thinking that puts children off maths. You can't move the goalposts halfway through a question. It's not badly a badly worded question, it's an incompetent question setter.
The problem isn't that the word has multiple meanings, it's that the same instance of the word "inside" is having different meanings depending on the subquestion. Since it is the same question and sentence for all the shapes, it should have the same meaning.
exactly, and I'd go as far as to say "present", "present" and "present" are 3 different words with the same representation (one is a express time, on is a verb and the last is a "object").
I think your justification about the usage of the term "inside" by comparing it with the word "present" is incorrect. The issue with "inside" in the question was that it meant two different things in ONE use of the word. Your comparison with "present" gives three different meanings in one sentence BUT only by utilizing THREE different uses of the word. You did NOT write "present" once and it had all 3 meanings, you had to write it three times to get those three meanings. Also the way you asked the questions to the AI already shows that you adjusted the original question to be easier to understand for the AI. You asked the AIs specifically separated each time - one question only for a circle and one question only for a triangle. If the original question would have been separated into 4 different questions (one for each shape) then the thought process would have most likely been different for the students than when it was put together in ONE question for all four shapes - especially with the circle being mentioned in the middle of the different shapes. To use the AIs as a comparison you would have had to ask them the question like this: "How many degrees respectively are inside a triangle, a rhombus, a circle and a pentagon?" And that could have then tripped up the AIs trying to use the same meaning of "inside" for all four shapes, too.
I tried asking Chatgpt and Gemini that question and ChatGPT answered "correct" while Gemini answered that circles don't have degrees since it's not a polygon
It still would not have tripped up the ai, but thats not because it understood the question. AI is basically a very complicated and huge prediction machine that predicts what should be the next word, based on how many times it appeared previously. So thus it would still connect the 2 questions separetly and give the right ans
As far as I'm concerned it's not really that the question has multiple interpretations but rather that it's just altogether malformed, like asking "how many meters are there in a kilogram?" - it's just gibberish. What does it even mean for a degree to be inside of a shape?
You are correct! I typed the question in exactly as written into ChatGPT and got this: Triangle: The sum of angles in a triangle is always 180 degrees. Rhombus: Each angle in a rhombus measures 90 degrees. Circle: A circle doesn't have angles; it's composed of curves. Pentagon: The sum of angles in a pentagon is 540 degrees.
You might complain that the question was poorly worded. But the real problem here was that the grader didn't understand the issue and marked the student's answer wrong rather than realizing that the student had answered the question as she understood the meaning of the word inside angle.
The grader DID understand the issue. If you answer the wrong question correctly, it still is the wrong answer. If an ATM asks you to input your PIN, and you input your phone's correct PIN (which hopefully is different to your bank cards), you wouldn't argue that your answer is technically correct when the ATM refuses to tell out cash, would you?
This is such an insightful remark. What is the purpose of grading and giving marks? Are you just checking boxes or are you trying to understand if the pupil is willing and capable of thinking critically?
English is not mathematics; the homework question is malformed -- the phrase "degrees inside" is mathematically undefined and itself wrong to use in a mathematical context.
@@professorhaystacks6606 The definition of shape is literally a closed 2D object. There are no non-closed shapes because it's in the definition that they're closed lmao
So the Daughter was marked down for either having too much knowledge or sadly lacking in psychic ability or the marker assumed she didn't know that a circle has 360 degrees despite knowing all the correct answers for every other question. In conclusion, the marker is either a robot or doesn't have the necessary intelligence to accurately mark mathematics exams.
Years ago, I took the Praxis exam. I quickly realized you could actually know too much and actually get a bad score. I realized that for many questions none of the answers were correct. But that one answer was what a teacher would give in a classroom if they didn't really understand the subject. By answering the questions that way, I got a max score on the test. Which means it wasn't a very valid test. 🤣
@sytherplayz Read what I wrote again. The reason the sum of interior angles is what you described, is because the sum of the COMPLIMENTS of the interior angles is 360 degrees.
I was once asked :- "Have you ever driven your car one handed?" My reply was :- "No! I've had two hands since birth". 🖐😎👍 It was a glib answer but unarguably a correct one, even though it wasn't the information they wanted. If a question is ambiguous then the any valid answer should be accepted as correct.
By saying “no” you’ve already given them the information they requested, haven’t you? You just also gave them extra information with the follow-up sentence.
@@divisix024 He didn't say he'd never driven a car _using_ only one hand (which was presumably the intent of the question), only that he'd never driven a car while _having_ only one hand. He certainly could have driven a car using only one hand while still having two.
@@divisix024 Not really! I often drive with only one hand on the wheel when changing gear but that again is not the info they were after. What they wanted to know, I would never admit to. Would I?
This reminds me of the story where two astronauts were on the moon and one bet the other that he couldn’t make donuts there. The other one jumped in the lunar rover and did donuts in the regolith. The other one said that THAT was not what he meant, the other replied that since the 1st one wasn’t specific, he (the second one) got to choose the definition of ‘donut’.
I generally agree that nonspecific questions should be accepting of any answer that technically fits, but the problem with people throwing homework questions online is that we are seeing them with zero context. For a student that has had a teacher telling them all week that circles have 360 degrees but, upon being asked, answers that a circle has infinite degrees deserves to have their answer marked incorrect. Maybe give them some kudos for a clever response, but they deserve to be marked off for not paying attention.
I’m with the daughter. The question was ambiguous. Mathematics is a science of precision, and so should its language be. Many years ago when I was in secondary school I was given a very challenging problem to solve involving the volume of a segment of a cone. The teacher had lifted the question from the national school leaving certificate exam but had changed one word, completely altering the meaning. She marked the test based on the original question and failed me because I answered the altered question. I eventually consulted with an expert mathematician who agreed with my answer. I never liked that teacher after that incident and I have never forgotten the incident over 50 years later.
Or, perhaps the class had just been taught about the '(n-2)*180' formula and the test was simply to find out if the kids could apply it correctly? Apparently, the mathematician's daughter could not.
@@undercoveragent9889 Applying the formula in reverse to the desired answer, you discover that the circle has four edges. Are you sure this is the correct explanation?
I think this would have been a great question to have asked _during a math class,_ so that students could discuss the ambiguity and gain deeper understanding. Tests shouldn't be ambiguous. They're for testing what you've already learned, not posing new questions you haven't learned about.
And by the logic of the circle, 360 should be the correct answer for ALL closed shapes! Either the question refers to degrees inside meaning sum of interior angles, or the absolute arc's angle, which always would be 360. And it isn't how presh explained with the present example, because it is using one generic instruction for 4 sub questions, so it would put the SAME meaning in all 4 If it doesn't then that is like a variable having 2 values you randomly take
@@TonyCAV8R The answer is right but the solution and the notation are wrong. Infinity is a limit. It can be a result but (almost) never a part of an equation. Especially at this level.
A circle is what defines angles. You don't need any corners. If you turn 360 degrees around, you will be right back to where you started, so there are 360 degrees inside any 2d object.
@@EaglePicking You could also argue there are an infinite number of angles inside a triangle and all but three of them are 180 degrees (the rest are located somewhere on the sides). Once you have decided you are not going to do that, and instead you are only going to sum the angles located at a vertex (as we did for triangle, rhombus, and pentagon) , then since a circle does not have any vertices it does make sense that the answer for the circle is zero. At least that is one reasonable interpretation of the question.
He didn't say it's ok. He says: "You would say you're using one word to mean two different things which should never happen in mathematical homework. This is a poorly phrased question. It should never be allowed." and I don't think the "different perspective" that follows contradicts this; it simply explains that it is a fact of every day life that you might come across ambiguity like this.
We are getting the question without context, but the student is not. The teacher undoubtedly taught what the answer was supposed to be in the special case of a circle. So the student rightfully loses those points for not paying attention.
Those were always the kind of question I were afraid of in math class because I know the teacher wasn't looking for the correct answer but for answer they thought was correct.
@@minor_2nd Ngl, this make me respect the curiculum in my country more. At the end of each test, student is allowed to recieve back their test paper and if there exist ambiguous questuon such as these, we are allowed to discuus it with the teachers. And we are also given extra make if the question is indeed ambiguous as a bonus question mark :D.
Heh, I still remember having boxes labelled 1st-7th and we had to write in the days of the week. I got the order correct, but started on Monday instead of Sunday, so each one was marked wrong. I maintain that the week starts on Monday. Justification: we call a Saturday and the following Sunday "the weekend", singular.
My immediate thought was "wouldn't the answer be 360° for all of them?" I think my instinct was to find the interpretation of the question that best fit all of the shapes, to which the inclusion of circle in the list limited most strongly. I don't imagine that would have gone over any better in the grading...
There should be no ambiguity in maths tests like that. Either the question as written should have been thrown out, or they should have accepted both 360 and infinity as answers. Even “undefined” should have been acceptable because relative to the other shapes, a circle is not a polygon so cannot have a sum of interior angles.
lol How on earth do you make (0-2)*180= infinity? The fact is, in order to construct a circle from an infinite number of pairs of intersecting lines, you would have to draw an infinite number of vertices... which is impossible. But guess what: we _can_ construct circles using just a compass. The circles _you_ imagine cannot even exist because there are precisely _zero_ straight lines along _any_ segment of any circumference of _any_ circle. By definition, the circumference is equidistant from the centre at every point but if there is _any_ straight line along the circumference, different points along that line will be at different distances relative to the centre. In any case; I have _never_ inscribed angles of infinity degrees when constructing any circle. There is a lot of intellectual snobbery in this comments section and it is purely for the purpose of justifying a mathematician's daughter who failed to apply (n-2)*180 correctly. Why are you all simping?
@@undercoveragent9889 hahaha! Not sure what’s worse, this snobbery you refer to, or massively overthinking things and coming out of the other end looking like a fool. Either way, appreciate the entertainment!
Arguably, a shape that is not a polygon can have a sum of interior angles. For example, a quarter disk (or, bluntly speaking, a slice from a pizza that has been cut into four equal slices) has three interior angles of each 90 degrees, and hence its sum of interior angles is 270 degrees. So I'd say that the sum of interior angles of a circle (or disk) does exist, and it is 0 degrees. By the way, not sure what the other replier is on about, but the (n-2)*180 formula only applies to polygons, i.e. shapes with straight (= non-curved) edges. It does not apply to shapes with curved edges (such as, for example, the quarter disk and the circle).
@@undercoveragent9889 It is not 360 degrees either, and that formula is not valid for any non-polygons (like a circle). But it just so happens that the circle is the limit of the regular n-gons as n goes to infinity, making 'infinity' (while a not strictly correct answer) a better answer than '360'
@@undercoveragent9889Even if practically impossible infinite vertices are mathematically possible. You also cant even draw a perfect square in real live should that invalidate the formula?
I don't think the question is fair. the purpose of a test/home work should never be to trick the test taker. The goal of a test/home work is to challenge the student while ensuring they learned the concept. When I asked chatgpt this question, here is what it said: Conclusion: The test is not entirely fair due to the inclusion of the circle. A circle does not fit the same criteria as the other shapes when discussing interior angles, which could confuse students. To improve fairness, the circle should be replaced with a polygon, such as a square or hexagon, or the question should be clarified to avoid ambiguity.
Unless, of course, the teacher spent all day telling the students exactly what the answer was for a special case like a circle. When people post homework questions online, we are viewing it entirely out of context. That student that was in the class was probably told how they were expected to do it and decided to not pay attention, thus, they get points marked off.
@@SgtSupaman And this is how we end up with Galileo Galilei under house arrest for the remainder of his life. Church told him the universe is geocentric, clearly he wasn't paying attention.
@@Eidako , wow, thanks for that wonderful example of an Appeal to Extremes fallacy. You do know a circle actually is 360 degrees, right? Don't get me wrong, a great teacher should absolutely give some kind of credit to a student that comes up with a clever response, but if they were taught the proper answer and can't supply it when asked, they aren't paying the attention they should be paying, which is an appropriate lesson in and of itself.
@@SgtSupaman There is no proper answer to this question - there are three quite valid responses to it. A circle is a 360 degree rotation, yes. However, a circle is not a polygon, meaning in the context of how many degrees are in a triangle, rhombus, etc., it has zero vertices and therefore there are zero (or "not applicable") angular degrees inside. Alternatively a circle can be considered to be a polygon with infinite vertices (the 3D modeller's approximation), in which case there are infinite angular degrees. The girl was rightfully confused by it.
I wonder how the teacher would describe the degrees inside a shape that is half of a hexagon on one side and half a circle on the other side? The logic applied to the circle should be the same for the other shapes. If the circle has 360 degrees then they should all be 360 degrees. I was wondering about 0 being correct as there are no discernible corners in a circle.
Another perspective: For polygons, we're looking at the sum of all interior non-straight angles. If we counted 180 degree angles, then all shapes would be infinite. As n approaches infinity for a regular n-gon, all interior angles approach 180 degrees. Would all interior angles for a circle be 180 degrees? If so, we exclude all of them, and get an answer of 0 degrees. As far as the fairness of the question goes, I'd rate it as not fair. It's an unrelated fact memorization problem tossed in the middle of measurement problems. It's also obnoxious that the circle breaks the sequence of shapes, 3-4-infinite-5. In instances such as this, all reasonably justifiable answers should be accepted. 360, infinity, 0, undefined.
The presents example is not the same as the homework question as each instance of the word present had a specific meaning and repeating the same word is fine, but in the homework question the word inside is used only once and the way it is meant to be interpreted inside has multiple meanings which is confusing and not standard mathematical practice.
I think its a fair question, but if the kid can justify infinity to their teacher, it should *absolutely* be considered a correct answer. 360 or infinity both make perfect sense, depending on the interpretation of the question.
Question is not fair because the structuring implied that the same question is being asked of all 4 shapes when in fact the circle question is different. Puttin the circle question in the middle of the other shapes sure makes it seem like you're asking the exact same question 4 times and not 2 different questions.
As a high school Geometry teacher, I would have given credit for the answer of infinity but used it to then have a discussion with the class so they understood what answer I was looking for.
i probably would have nocked down infinity. maybe partial credit. i would not have accepted 360. full credit would have gone to not defined. given it is elementary i could see a strong argument for zero as circles don't have interior angles therefore the sum of nothing should be nothing but it is in the end not correct. probably would give it half credit.
The best mathematicians-to-be in the class will say infinite. The best test takers (often times the top of the class) will say 360. The best logisticians in the class will say both. You're right... it's a poorly constructed question. Deep down, I'm compelled to say infinite, because you can prove it logically.
The real problem is that an answer without a reason was expected. The reasoning behind an answer, in school, should hold more value because you are trying to get students to actually think. A far better test would ask for ‘why?’, in which case both answers would have been acceptable. A “correct” answer with the reasoning being “Because that’s what I memorized” should only get partial credit.
Or, perhaps the class had just been taught about the '(n-2)*180' formula and the test was simply to find out if the kids could apply it correctly? Apparently, the mathematician's daughter could not.
@@undercoveragent9889 The (n-2)*180 formula only applies to polygons, which are shapes with straight edges. A circle is not a polygon. That being said, the daughter did give an answer that corresponds to the (n-2)*180 formula if a circle is considered a regular "infinite-gon".
@@undercoveragent9889 You appear to be arguing that a circle has four sides: if (n-2)*180 = 360, then n=4 If that is the purpose of the question, then it's not just a bad question; it's a terrible one.
Assuming _internal_ angles are meant, it's 11) 180°, 12) 360°, 13) N/A, 14) 540°. A polygon with n = ∞ isn't a circle, it's a polygon with an infinite number of sides. Circles don't have sides, so they don't have internal angles. "Inside" would have to mean different things depending on the shape for a circle to have 360°.
The problem is that a degree isn't a thing on its own. It's a unit of measurement. An equivalent question would be asking how many centimeters are inside a shape but for some shapes asking about the circumference while for the circle asking about the diameter.
Infinity is correct within the context of the other shapes. If the teacher forgot about this, both answers should be correct. If the teacher deliberately meant it as a trick question, he shouldn't be a teacher anymore.
I stopped worrying about what my teachers BELIEVED in 3rd grade. Math and science were presented at such a low level that it wasn't until college that I needed to "study" them at all.
Science has specific definitions to avoid ambiguity. "Velocity" in science is a speed and direction, a vector. In English, it can be the same as speed. "Speed" is a scalar in science with no direction. It is not a vector.
Ah yes, the famously unambiguous maths. Where sin^2(x) and sin^-1(x) mean completely different things, where A’ can be a transformation or a derivative, and where the unwritten logarithmic base is always 10 except when it’s 2.
one of the mistakes I made in the past was about ambiguous questions, like the relative position of two lines, I used the question 'what can we say about the two lines (d1) and (d2), until one of the students wrote: 'they are beautiful' , the answer stunned me and I thought: 'this can be an answer if he sees them like that' and I gave him the full mark and never repeated an ambiguous question again.
Planar polygons are not real objects; they're mathematical constructs. The answer depends upon the arbitrary number that you choose to represent one revolution in your system. 360 was chosen by some recent inhabitants of Earth, others chose two (2*pi) or 100. It could equally well be one (1). Martians could have a system based on their ~670 local day orbit.
@@__christopher__ 2*pi has never been used for degrees. it is used for radians. i'd argue that the way degrees are traditionally defined the polygons would have degrees but a circle in fact would have to be considering the central angle which is more properly defined in radians then degrees.
I have another fun method to solve for the sum of interior angles. Since, the sum of interior angles can be given by 180°(n - 2), where n is the number of sides. Instead of taking n as infinity we can also take n = 0 for a circle. Now when you subsititute the value in the equation you get the sum of interior angles in a circle to be -360°. It seems absurd at first glance but if you think about it, the negative sign actually refers to the measure of central angle instead of sum of interior angles. We can also play around with the values of n to verify the logic of negative sign being the measure of central angle. For example if n = 1, which would refer to a straight line, where the equation gives the sum of interior angles to be -180° or measure of central angle to be 180°. Again if n=2, then the structure will be for any two lines joined together at a point, where the sum of interior angles is 0°. Which for me it again makes intuitive sense because the structure is not well defined. So even if we do not know that the measure of central angle in a circle is 360°, we can conclude the same using the equation. Mathematically speaking I would argue that the question was not poorly phrased because if it were then the equation should not have come to the same conclusion.
true, as you said that a word can have multiple meanings in sentence. but as you said right before it a word in a mathematical question should *never* have multiple meanings
Your graphics are top notch, and track seamlessly with your narrative - begging the question “Are your narratives top notch?” Well duh, yes your narratives are top notch also. Thank you for your time, effort and passion. You are endlessly entertaining and thought provoking. I was briefly a math teach after leaving the U.S. Navy (as a civil engineering officer), I taught high school algebra and geometry. Whilst going through high school (like most of my peers) I disliked word problems. Teaching algebra and geometry led me to the realization that word problems are the superior method of imparting wisdom and analytical thinking. Your videos are a wonderful expression of both wisdom and analytical thinking and if i were still teaching HS subjects I would be discussing algebra and geometry using your videos as much as possible. Bless you and Bravo Zulu on your efforts and achievements.
If there are 360 degrees in a circle, then there are 360 degrees in all those other shapes as well, as we defined "degrees in a shape" to be "degrees around the center" rather than "degrees of the angles between the sides"
Though once you get hexagons and higher, you get shapes where there is no point which can "see" the entire interior of the shape, which makes "center" a murkier concept. I suppose you could phrase the concept in terms of maximum winding number...
You could argue a circle is 360° because a circle is made of two half circles, which each have 2 angles of about 90 degrees. So 90° x 2(amount of angles per half circle) x 2(amount of half circles per circle) = 360°
*I don't agree with that perspective: **9:04**.* We don't have a "single word with 3 different meanings", we have "3 different words that are written the same", at least that's how I see it. For your example to work, you had to write the word "present" 3 times, that didn't occur in the polygons question, the word "inside" was written just one time. So, for me, if I have one question to be answered, we should apply the same logic to all of its items
There are no straight lines inside a circle. The sum of the angles could be interpreted as 0 or infinity. That would mean it is undefined. If you’re looking for the central angle, then yes, 360 degrees. Then you could logically argue that all the shapes could have a central angle of 360 degrees.
this was a rollercoaster of a video, i was initially annoyed at the teacher, but then your explanation about the English language and how the meaning can be inferred, that was beautiful. Is it weird, that i felt moved with how nice this video is. purely logical explanation and by the end of it there is no finger pointing.
If the person asking the question knows that there are multiple ways to interpret an angle being "inside" a shape, then the question should have specified which kind of measurement was intended. If the desired answer involves using multiple distinct interpretations of "inside", then clearly the author has multiple interpretations in mind, and therefore the question was deliberately ambiguous, and a deliberately ambiguous question does not have just one right answer.
Before watching this video I recall that the sum of the (inside) corner angles of a triangle is 180°. like a 60-60-60 triangle with identical angles and side lengths. For every (inside) corner you add to the polygon I think you add a fixed number of degrees. Maybe 180° for every extra corner. Because a square has 4 x 90° = 360° (180° more than a triangle. Thus: 11) Triangle = 180° 12) Rhombus = 4 corners = 180° + 180° = 360° 13) Circle = zero or ∞ corners (maybe 0° or ∞°). My answer is Ø (null) 14) Pentagon = 5 corners = 180°+ 2(180°) = 540° EDIT: After watching most of the video I agree that "how many degrees inside", as stated in the original question should be interpreted as "the sum of interior angles". As there are no actual angles or corners along the line of a circle, 0 may be considered the most "correct" answer.
Fun fact: How to tell if a point is inside a closed figure, and how to orient handedness of the figure (in 3D) Choose any point P on the plane. Choose any point S on the figure. Define a vector V = P - S. Define A = arg(V) => the angle of V wrt any origin. Integrate A over a closed loop path of S once around the figure. If A is outside the figure, Int(A) = 0. If A is inside the figure, Int(A) = +/- 360 degrees. The sign of Int(A) depends on the direction of the path integral and so gives a handedness orientation By convention, right handed equals a positive integral for a counter-clockwise traversal (from x-axis towards y-axis yields positive z-axis). Since the choice of origin was arbitrary but does not affect the result, the result is invariant wrt translation and rotation.
The other way to avoid ambiguity is to draw in straight lines from the vertices of each polygon to its geometrical centre in a radial type way. The angles, therefore, at the centre point would always be 360° in any shape or, in other words, the central angle will always be a circle around the geometrical central point, - so that could refer to “angles total inside a shape“ in all instances. Other than that, the ambiguity raised in the video was a good one. The girl’s response of infinity for the circle was a valid one based on the reasoning given and it should’ve been clearly stated what they were referring to when asking the question! A very good video and very informative. Excellent.🌞
My immediate thought was that the question was meaningless, and one needed to re-write the question in a meaningful way before answering it. It is not clear why the vague and almost meaningless phrase "how many degrees inside" should be translated as "what is the sum of the interior angles expressed in degrees of."
No, that wasn't the question. Clearly, this was a geometry class for kids where they were being taught how shapes can be constructed. The formula '(n-2)*180' computes the sum of all the angles that have to be inscribed in order to construct shapes where 'n' is equal to the number of pairs of intersecting lines or, 'the number of vertices'. Or, keeping it related to trig: the question refers to the minimum number of triangles are required to create regular polygons. It takes one triangle to make a triangle, two to make a square, three to make a pentagon, ... and each triangle adds another 180° to the sum of all angles that must be inscribed in order to construct the shape. You cannot construct a triangle without inscribing three angles that sum to 180; you can't construct a square without inscribing angles that sum to 360 and you cannot construct a circle without inscribing an angle of 360°. The simple application of (n-2)*180 holds in every case unless of course you are the daughter of a mathematician when '(0-2)*180' somehow equals infinity.
@@undercoveragent9889 how can you correctly state that this formula by definition is applicable only to regular polygons and also fail to see that circle is not a regular polygon, making the formula not applicable?
@@undercoveragent9889This is so true. A circle has 4 sides as given by the formula. We should question nothing and believe whatever goes in our ears🎉🎉🎉
@@undercoveragent9889 the formula is very specifically defined for polygons only and requires significant constraints to work for concave polygons at that. it is not defined for non polynomials such as any shape with a curved edge like a circle.
There are no internal angles in a circle, and this question implies internal angles, so the answer is N/A. I would also score 0, 360 or infinity as “correct”.
Depending on the grade of the student and with the question being an "easy" one here (with the triangle, rhombus, and pentagon), then it is obvious the answer the teacher was looking for is 360 degrees. The student can most likely bring up the reasoning and ask the teacher for the point back, but if this was a real final exam, then there would have been given a point for infinity, as student do sometimes see answers that the question makers dont. One example I have seen is (do note the question isn't originally in English): You have a roulette wheel with 2 labels of "spin again", 1 label of "10% off", 1 label of "20% off" and 2 labels of "nothing", to determine the amount off you get from your purchase. What is the chance you get "nothing"? What is the chance you get "20% off" ? The answer key said: 2/6 or 1/3 for the "nothing" label, while 1/6 for the "20% off". But some student submitted answers of "1/2" or "2/4" for the nothing, and 1/4 for the "20% off". I can see the official gradting notes does indeed state that those student answers were accepted as correct even though it didn't match the official grading key.
The only correct answer is "What do you mean 'degrees inside'?" My teachers hated me for that, but indeed that's always the right answer when the question is poorly phrased. EDIT: also, what if there was a circle or a square inside the triangle? What would that mean? How many angles would it have inside? How does phrasing the question like that helps kids learn when they haven't developed this kind of subtletly in their thinking yet (or even teenagers)? Except if the teacher has got a plan to make them think through it and then make a point out of it later, rather than simply grading it as "right or wrong", that's really harming more than helping.
I would say the daughter is more educated than the teacher... she took the meaning of inside saw that majority fit this definition and used that definition to come up with the answer for circle... that's how math started... you establish a rule and then solve stuff according to the given rule...
Or, you could break it down by cases, and answer, "If by 'degrees inside a triangle' you mean the sum of the triangle's interior angles, the answer is 180. If, however, you mean the sum of the angles subtended by the sides of the triangle with respect to any point in the interior of the triangle taken as a vertex, then then answer is 360," and continue in that vein.
@RolandHutchinson It is a good option and is usually what I do as an adult in conversation (not usually when answering written questions on exams because I expect those to be properly phrased by default, unless the teacher is willing to make a point out of it and make students think, but they usually don't, in my experience. I don't know what this particular teacher was going for, so I can't speak about this particular case). I don't think middle schoolers or even high schoolers usually have that level of mental flexibility or communication skills, though, even the smarter ones, especially because that's not usually how subjects are taught at school. Answers are either this or that most of the time and most teachers don't welcome lateral thinking and exploration very well (again, not the ones I was taught by, even in college). I certainly wouldn't have come up with that answer at that age, and I was thinking more on that vein, and I don't think unclear (exam/homework) questions deserve elaborate answers either way. "Real", adult life is a different thing. If you're trying to sort things out with someone on your job or personal life, it is not necessary to be nitpicky, and it is better to elaborate and make sure everyone is on the same page.
I posed this question to ChatGPT - Give the number of degrees inside each of the following figures - (a) triangle, (b) rhombus, (c) circle, and (d) pentagon. Its answers were - Sure, here are the number of degrees inside each of the figures: (a) Triangle: The sum of the interior angles of a triangle is always 180 degrees. (b) Rhombus: A rhombus has four interior angles, each of which measures 90 degrees. So, the total number of degrees inside a rhombus is 360 degrees. (c) Circle: A circle does not have interior angles in the same way polygons do. Instead, it's measured in terms of its central angle. A full circle has 360 degrees. (d) Pentagon: The sum of the interior angles of a pentagon is given by the formula (n-2) * 180, where n is the number of sides. For a pentagon (5 sides), the sum of interior angles is (5-2) * 180 = 3 * 180 = 540 degrees.
Then define 'quadrant'. If there are no degrees in a full circle then the number of degrees in a quadrant would be 0/4, wouldn't it? Maybe you have misunderstood something very fundamental?
@@undercoveragent9889 Circle: a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center). Quadrant: each of four quarters of a circle. We are talking about different things here... The degrees of the inside angles of a polynomial are not the same as degrees of arc of a circle.
@@Golfnut_2099 "Circle: a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center)." Yes and further, the circumference inscribes an angle of 360° around its centre. Given the context of the geometry lesson being taught to these kids and that they were being taught how to construct shapes accurately using only a pencil, a compass, a protractor and a ruler, this 'test question' appears to be aimed at testing how well the students apply the formula (n-2)*180. In essence, all of those defending the mathematician's daughter are suggesting that the formula does not hold for 'n=0'. Maybe the question should have read: "What is the sum of all angles that must be inscribed in order to construct the following shapes?" But, I think that from these kids' point of view, that would simply be a paraphrasing of: "How many degrees in the following shapes?" The fact is: you cannot construct a circle without inscribing an angle of 360°. Sadly, Presh missed an interesting implication of how circles and triangles are related. In fact, you _could_ imagine the centre of the circle as being the 'apex' of a flat cone and if you cut the cone down its centre, you get a triangle. Right? You _could_ think of a circle as being a line that is parallel to a line through its centre and perpendicular to its plane and what you have is _two_ lines that do not intersect. Quadrant, a 90° section of a circle. What about 'sextant'? Any relation to 60° at all or are we going to continue along the disingenuous path? If there are 60° in a sextant, 90° in a quadrant then in what way is there _not_ 360° in a full circle?
@@undercoveragent9889 but if that is how we are defining "degrees in a" then we should be looking at the central angle from the center of each shape to be consistent and every shape listed would have 360 degrees inside it.
my answer would be zero, because the inside wall of a circle is a curve, so there are no points of intersection of straight lines to measure interior angles.
I probably would’ve given both answers in that situation. I’ve been doing that and leaving a little note next to the question my whole life explaining the ambiguity of the question.
triangle has 160deg = half rotation square has 360deg = full rotation from there its easy to remember, that every additional corner adds +160deg, adds a half rotation -> anglesum = (cornerount-2)*160deg circle is degenerate/asymptotic/undefined case, where infinite corners are 2+infinite half-rotations, which is nonsense. A more reasonable substitution is "no corners means; 0 MINUS 2 half rotations == -2 half rotations, to at least evade infinity, the sum of 0 corners in a triangle adds up to MINUS 360degrees. This is mostly fine in the way that spinors work, or how you just rotate in the opposite direction, or in how you can just abs() the result, ignoring the sign.
His daughter should have raised her hand and asked for clarification. I always did this, and it "saved" me from reduced marks and later frustration. As a bonus, it's nice to see the faces of teachers when they realize they messed up, after you point out the inconsistency.
I haven't watched the video, but the answer for any convex polygon would be n * 180 - 360 = (n - 2) * 180, where n is the number of sides. This is obtained by putting a point inside the polygon, and then drawing rays from that point to each of the vertices. You will end up with n triangles, each of which has 180 degrees inside it. We then need to subtract the angles where they all meet at the point inside the polygon, which add up to 360 degrees. So, for a triangle, we have 1 * 180 = 180, a quadrilateral would be 2 * 180 = 360, a pentagon would be 3 * 180 = 540, and a circle would be infinite.
Answer for a circle is 0. The answer is the sum of interior angles. These points are corners in the shape. Since a circle has no corners, there are no angles to add up and thus the answer is zero.
No, you can divide a circle into as many angles as you wish from the center. The sum of them will always be 360 degrees. Of course, this is true for ANY 2d object.
My friend, sadly gone from us, who was a games programmer in the 1980's decided it was more practical to have 256 degrees in a circle. He built sine and cos tables accordingly and it made his life easier becase of the 8-bit wrap.
@@undercoveragent9889 You clearly don't understand the issue at hand and you seem to think the almighty Google is the world's superior problem solver. I suppose even the Fields medalist and his daughter fold at the hands of almighty Google.
@@undercoveragent9889 Here is a shocker for you. Google "How many stars are in the Big Dipper?" You will find 7. That is incorrect. The correct answer is 8, as Mizar and Alcor make up a double star. Google is not a reliable source of information. All that Google does is take your search query and find another source that it thinks is relevant. You can publish absolutely anything you want online and, if Google thinks it is the most relevant result for a search, Google will present that information first without any sort of verification. Funnily enough, even searching "does google verify information" will not give you what you are looking for. It gives results for how Google verifies _identity_ for accounts, not information presented after a search.
"How many degrees inside the following shapes?" 360, for each of them. Consider pizza slices from a pizza. If we sum the angles of the tips of each slice, the sum will be 360 degrees, regardless of the (convex) shape of the original pizza. (Note: the word "inside" in the question implies that the tip vertices of all slices correspond to some arbitrary "central" reference point _within_ (= not _on_ ) the perimeter of the original pizza. However, if the tip point of the slices could have been chosen to be _on_ the perimeter of the pizza, then the sum of angles could have been 180 degrees instead of 360 degrees.) However, if the question meant the sum of the inner angles at the _vertices_ of each shape, then the correct answers are: 11) triangle: 180 12) rhombus: 360 13) circle: 0 (and not "infinity"; a circle doesn't have any vertices) 14) pentagon: 540
1) A circle is not a polygon, so "infinity" is wrong. Adding angles forever will still never create a circle. Think of it this way: The limit of the polygon approaches a circle, but what does the limit of one of the interior angles approach? 180°. So if you were able to reach the limit, you would simultaneously have a circle and a straight line. At the limit, the shape becomes undefined. 2) Degrees are degrees, they are not different meanings. An interior angle of a polygon is a measure of arc swept by those arms. Circles are 360 degrees of arc. 3) The question is not ambiguous. The primary lesson is about understanding the relationship between circles and polygons. 4) Most fundamentally, if a circle is anything other than 360°, then measuring angles becomes meaningless. The degrees of angles are what they are BECAUSE a circle is 360 of them.
The exterior angles of a circle can be argued to add up to 360 degrees, but not the interior ones. The question specifically mentions interior angles. For interior angles, we can either argue that a circle is a limit as a polygon approaches infinite vertices, in which case, you have an infinite sum of numbers infinitesimally smaller than 180 degrees, or that the sum of interior angles is 0 degrees, since circles don't have any vertices (and note that for normal polygons, we're not counting any of the 180 degree angles that we get when we pick some random point somewhere along a side. Only vertices count!).
@@Keldor314 Your first two sentences are false. A circle has no angles, a circle is made of arc. A circle is 360° of arc, or the whole set of points in a plane equally distant to a given point. The question specifically says "degrees", not "interior angles". "Angles" is not in the problem at all. A circle is not a polygon. If you can't grasp this fundamental idea you will never understand the problem. A near-infinite sided polygon is an approximation of a circle, useful for some mathematical approximations, but it is not a circle. If you say "circle" then you are not talking about an infinite sided polygon. These are two distinct things with different properties. Circles have no angles. Circles have 360 degrees of arc. It is degrees of arc that you are measuring when you measure an angle inside a polygon. The number of degrees in that angle measurement is a direct reference to the 360 degrees of arc in a circle. The whole point of this problem is to highlight the relationship between polygons and circles. Most of what I have said here is just a repeat of what I said above, you just need to read carefully and think about it. And read the original problem again.
@@TashiRogo Alright, but none of the polygons have any degrees of arc at all, not being constructed out of arcs, so clearly the question is conflating two poorly matched constructs. So should we add a refrigerator to the question so we can add the degrees Farenheit to the mix? Or maybe the one degree of freedom the refrigerator door hinge has? The problem with mixing interior angles with circles in this way is that we've deftly replaced one measure by its counterpart. Imagine you take the vertices of a polygon and replace them with arcs, "rounding off the corners", so to speak, so that you're left with a smooth curve. How many degrees are inside of it? Well, 360, of course, since you add up the lengths of the arcs after normalizing them (and subtract ones curving "outward"). But this doesn't match the amount of degrees it had before unless it started as a quadrilateral! What happened? Well, those arcs correspond to the *exterior* angles of the polygon, not the interior ones. The fault lies with the choice of defining a straight line as 180 degrees and a sharp turnaround as 0. They're backwards. So I guess I agree with you about the inside angle of a circle. It was the polygon examples that suffer from unnatural definitions of angles. I suppose we should measure angles as if we're bending a wire instead of unfolding a piece of paper. How did that folded over piece of paper get considered the starting point anyway?
@@Keldor314 They aren't poorly matched constructs! 😆 The whole point of the problem is to show how polygons and circles are related to each other. Circles have a whole set of properties that make them useful for solving various kinds of problems. Polygons have a different set of properties that can be used. Degrees can be applied to both, and by using this relationship, you can combine the various properties of each together to create a bigger tool-set. It's easy to take for granted what it means when we label angles with some number of degrees. Lets take a square for example. It has four 90 degree angles, but what does that mean? 90 degrees of what? What it means is that if you drew a circle centered on a vertex of that square, 90° of that circle would be inside the square. It's the same for any angle of any polygon. The degrees of the angle is a measure of how much of the circle that is centered on the vertex of that angle would be inside the angle. So, although polygons contain no arc, and circles contain no angles, the angles in a polygon are measured in degrees of arc, and the degrees of arc in a circle are demarked using a central angle. Also, the central angle only works for circles because they are completely uniform with respect to their central point. The central angle of a polygon is a nonsense concept. It doesn't measure anything useful. If you wanted to measure a polygon where you have rounded the corners somehow, you would have to break it down into smaller parts and use multiple measurement concepts to accomplish this. Once you understand this, you can see how silly it is for all of these people to say that the student's answer of infinity degrees for a circle was a good answer, or how zero is an acceptable answer. The degrees in the circle are what is used to measure the degrees in the polygons. If you say the degrees of a circle is not 360, then saying that a triangle contains 180 degrees doesn't even mean anything.
and a spherical triangle has 270 degrees total interior ( 3 @ 90) start at the north pole, head south to the equator, turn left 90 degrees, travel 90 degrees along the equator, stop, turn left 90 degrees and head back north to the pole and you find yourself facing 90 degrees ( perpendicular) to your original path thus 3 points of 90 degrees each making it 3X90 = 270
The sum of supplementary angles of any shape is always 360°. If you have a triangle that would give three angles a1, a2 and a3. For each angle, you'll get 180-a1+180-a2+180-a3=360 which gives a1+a2+a3=180. You can generalize : for an x angles shape, you'll get sum (1 to x, 180-a(x)) =360 Which give sum(1 to x), a(x)) = 180 (x-2) Also said, the sum of a x sides shape angles is 180*(x-2) A circle can be considered as an infinite sides shape thus having a sum of interior angles of infinity.
I agree, as the question is stated 'Infinity' is the correct answer ... ..to any point on a circle from the center u can turn a maximum of 360 degrees to go to any poin on the surface, however thats not the only angles 'inside a circle', we can do this two ways... by deviding it into n triangle slices aka n-gon, [180(n-2)] deg aproaces infinity as n increases... of adding angles between points on the surface of the circle..and as ther is infinity many points thers also infinity many angles... the sum of angles goes to infinity... the only matematically correct answer is Infinity... ..Physically u can argue the smallest distance between points on the edge would be The planks length as u cant pack infinite atoms together... (about 1 * 10^-35 meters) ...wich produces an n-gon with 10^35 surfaces .. so the maximum namber of physically possibly angles would be 180( (360+1)/10^-35 ) - 2) =180(361 * 10^35) - 2) degrees, but it higly depend on whats atoms the circle r made of .. differen atom can be packed differently and depends of energy(temp) and external Pressure) .. or about 6.497999999999999999999999999999999999998 × 10^39 degrees is physically possible, 180(361 * (1/Plancs length) ) - 2) is a more pricise anwer corrction: the angles (using planks length) adds to: 6.49799999999999999999999999999999999964 × 10^39 degrees... (in addition: the smalles orbial width of an electron been measered to about 1*10^-22 meters, but nera absolute zero and with high enugh pressure it aproaches 1*10^-35 matematically)
I've had tests to work at nuclear power plants which have questions just as poorly worded. In one case, it was a 10-question test which you had to pass with at least 80% (8 questions correct). One of the questions was extremely poorly worded so I asked what they meant. However, the proctors said they couldn't talk to me during it. I had just to answer by guessing what they meant to say. After I finished, I went to the overall instructor to explain why I was so frustrated. I indicated what question I hated, and he agreed that the question stunk. If someone missed it, they would have only had 1 more question they could miss without failing.
The question could have been stated more precisely as "What is the sum of the interior angles in the following shapes?" . If "degrees inside a shape" is taken to mean the sum of interior angles inside the shape, that meaning should hold good for all shapes including the circle for which the correct answer then would be infinity. The example with "present" used three times is a false analogy because the word "present" is used thrice in different contexts. But in the math question, the "degrees inside a shape" although constituting loose language is used just once in the same sense for all shapes. So if the answer for circle is expected to be 360 degrees, it should be 360 degrees for all shapes because then it can be concluded that "degrees inside a shape" is not used in the sense of "sum of interior angles", or else the answers should be 180 degrees, 360 degrees, 540 degrees, and infinity degrees respectively for the triangle, rhombus, pentagon, and the circle using the 180(n-2) formula.
I could be wrong here, but I've always felt that a curve and an infinite number of points approximating a curve are not the same thing. A circle has no interior angles, so why are we trying to add them up? Now the question didn't ask to add up all the interior angles, but it also didn't bother to use proper grammar. "How many degrees inside the following shapes?" Presumable what they meant to write was, 'How many degrees ARE inside the following shapes?'. Since we don't have an objective question to answer, what are all the possible questions that could have been meant by this question (I've tried to list a few): a) Using the formula 180 degrees (n - 2) = a, where n is the number of sides of a given shape. What is the value of a for each of these shapes? In this case the answer depends on how many sides you consider a circle to have. If you're taking a purely mathematical approach, a circle has 0 sides, and therefore the answer should be -360 degrees. Otherwise you could take a more philosophical approach in which you could argue for any number of sides. The most obvious arguments being that a circle has either one continuous side or infinitely many sides reaching answers of 180 degrees and infinite degrees respectively. b) What is the sum of all interior angles of each shape? In this case the answer depends on how many angles a circle has. As before you can argue for both infinite angles and zero angles, and in the case of infinite angles the answer is the same. However, in the case of zero angles we now reach an answer of 0 degrees. c) What is the central angle of the following shapes? Now the answer for each shape is 360 degrees. d) How many degrees are inside the following shapes? And now, the answer starts to get real tricky. Standard test-taking practices suggest that if we are going to interpret this question, we need to interpret this question the same way for each shape, although there is no law of the universe that requires us to do this. We could, for example, interpret point a) for the triangle, rhombus, and pentagon, and choose to interpret point c) for the circle. This would give us the answer the school was expecting, but why would you ever assume this is the interpretation the school wanted. Well, when you've gone to school for as long as I have, you just get a sixth sense for this sort of thing. Or there's my favourite interpretation of this question which is that there are an infinite number of points within each of these shapes and each point has 360 degrees because of course it does, and therefore each shape has infinitely many degrees inside of them because you really should be more precise when wording a test question. Oh and I also forgot to point out that for options a, b, and d; you can also logically conclude that the answer for a circle is undefined, nonsensical, or meaningless. For example, if you were interpreting the question as use the formula for calculating the sum of all interior angles of a polygon, you could then say that a circle is not a polygon and therefore an answer does not exist or make sense, etc. But basically, if I had to sum up my opinion: the school is wrong, and this shouldn't be surprising.
I suggest the answer is "undefined", as a circle does not have interior angles the way a polygon does. The central angle is 360°, yes, but that's not the same as a polygon. I guess I agree with ChatGPT?
The answer zero can also be defended. We don't sum all the 180 deg angles along the straight edges of the polygons, we only aim the angles where straight sides meet at some other angle other than 180. Two arguments follow on from this point There are no straight sides on a circle, so there are no angles to sum. The sun of zero of anything is zero. Alternatively, if we are going to allow "infinity" as the number of "sides" of a circle, that means the internal angle "is" 180 and should be ignored, as with a non-angular node anywhere along the sides of a polygon. We should therefore ignore all these "infinite" numbers of angles. To be more rigourous, the above argument should be rephrased in terms of the limit as n -> infinity. But that's not how the problem was set ...
A third aspect to consider is that a limit _cannot be used_ to determine the sum of interior angles of the circle. For example, we could define a sequence of polygons such that their shape approaches that of a triangle as the number of vertices approaches infinity, but where the sum of interior angles is and remains, say, 360 degrees, _even in the limit_ . [ Start with four vertices A = (2,2), B = (0,2), C = (0,1), D = (1,1). At the k'th iteration, add the vertices Pk = (0, (1/2)^k) and Qk = ((1/2)^k, (1/2)^k). Then the polygons ABCD ABC(P1, Q1)D ABC(P1, P2, Q2, Q1)D ABC(P1, P2, P3, Q3, Q2, Q1)D ... ABC(P1, P2, ..., Pn, Qn, ..., Q2, Q1)D etc. are all quadrilaterals, each with a sum of interior angles that equals 360 degrees, and hence the limit of the sum of interior angles (as n --> infinity) is also 360 degrees. However, as n approaches infinity, the shape of the polygon approaches that of the triangle ABO (where O = (0,0) is the origin). ] Would that then mean that the sum of interior angles in a triangle is 360 degrees? No, of course not -- the sum of interior angles in a triangle is always 180 degrees. This shows that the limit of the sum of interior angles generally cannot be used to determine the sum of interior angles of the limit shape.
@@yurenchu Your proposed construction has a problem with treating the vertices inconsistently - it counts them all for "number of vertices" but only counts four of them for determining angles. The point - that the limit of a property isn't the same as the corresponding property of the limit - is valid, but the example isn't entirely convincing. It's less directly related, but I like to use the staircase construction of a diagonal line to establish the principle - a staircase connecting (0,0) and (1,1) has length 2 even as you double the number of steps, but converges pointwise on the straight line connecting the two points, which doesn't have length 2.
The issue that I see is that when questions are grouped together like this, that usually means there's a common trend, or they're all very similar. Randomly throwing a curveball about radians and center points isn't only a (slightly) different level of work, but completely differs from the rest of the question!
I would have answered infinity or zero for the circle, depending if you consider a circle having zero or infinite corners. "Studies show that girls and boys with exceptional grades are most likely the teachers favorites." I don't know the exact original sentence, it was in German anyway, but there is a part where a Mathmatician should want to add brackets. It could mean "(girls and boys) with exceptional grades" or "girls and (boys with exceptional grades)". Math is inherently based on logic and should be based on logic always, whereas language is inherently based on chaos and nearly void of logic. That's why the word "infinite" is pronounced different from "in finite". Language does not follow logic, so Math should not follow language like that. That being said, I had a similar experience in Math once. I was asked to prove n*(n/(n^2-1))^(1/2) = (n+n/(n^2-1))^(1/2), given multiple examples. Thing is, in the examples there is no plus sign on the right side, which I completely forgot about. Basically, if you have a natural number ahead of a fraction, in concrete digits, it means they are added, but with variables, it means they are multiplied.
There is a single, generalized definition that works for all cases and yields the expected results: Follow the outline tangentially. Whenever you reach a corner (i.e., a single point where the path is not differentiable because the left limit is not equal to the right limit), make an instant counterclockwise turn from the previous tangent until you align with the new tangent. This way, you will have turned 180(n-2) degrees traversing a n-gon and 360 degrees traversing a circle, and you can figure out degrees for even more complex shapes.
When I try following those instructions, I get a rotation of 360 degrees for all polygons. For example, with an equilateral triangle, I turn 120 degrees at each corner in order to get from one side to the next.
The question is worded to be technically correct only for the circle, for polygons you have to use your intuition to realize the intended goal is an interior angle sum. Somewhat ironically if anything the questions the dude's daughter got right should be thrown out and the one she got wrong kept in, if you take things super literally anyway..
If the answer to the circle is 360 then by the same logic all shapes will also be 360
Ding ding ding! We have a winner. There are always 360 degrees inside any 2D plane, therefore they are there inside any 2D object.
Agreed, with the caveat that any point exactly on a side or corner of a 2D object will necessarily have fewer degrees (since in at least some of the directions from that point, there is no 2D object).
A circle is not a polygon. It's true that the question is kinda vague. I would answer it so that the circle is 360 and the rest of them, polygons that they are, follow the formula: sum of interior angles = (n-2)*180 where n is the number of sides. If I got docked, I'd complain to the instructor that his test questions are vague.
that was my first thought, a circle is 360, everything should be 360, except i knew a triangle was 180, so i was stuck.
That's called external angles. Even works for non-convex polygons/shapes if you allow negative angles for the concave parts.
This is another problem where the controversy between answers can be traced to the ambiguity in the initial question.
It seems like people ask ambiguous questions when they lack understanding of the subject they’re asking about.
there are so many problems like that now a days. weve all seen the ones on instagram and tiktok that are something like what is 2 / 4(5-3) but it uses the old divide symbol and doesnt say where the divide actually is. so the answer is both of them it just depends on where you put the brackets. theyre awfully worded and constructed. its mainly just to get engagement
The teacher just want the answer they want (Or the answer they can easily understand), not the deep discussion we are having here.
@@alykadane7206 Problem is, then you've got teachers marking answers wrong because they know less about the subject matter than the kids they're "teaching." I had (mostly) fantastic math teachers back when I was in school, but it definitely seems like quality has fallen off a cliff in the past couple decades.
Infinity was the correct answer. 360° was wrong. Unless you think someone has just proved that a circle is a four sided shape. Or, if you interpret the question differently and it's about the number of degrees around the centre, then all 4 shapes have 360°.
However you interpret the question you have to apply it to all the shapes the same manner. It is either about internal angles or it is about the shape being closed. This is just another example of the person setting the question having insufficient knowledge about the subject. It's the kind of low quality teaching and wooly thinking that puts children off maths. You can't move the goalposts halfway through a question.
It's not badly a badly worded question, it's an incompetent question setter.
The problem isn't that the word has multiple meanings, it's that the same instance of the word "inside" is having different meanings depending on the subquestion. Since it is the same question and sentence for all the shapes, it should have the same meaning.
exactly, and I'd go as far as to say "present", "present" and "present" are 3 different words with the same representation (one is a express time, on is a verb and the last is a "object").
I think your justification about the usage of the term "inside" by comparing it with the word "present" is incorrect.
The issue with "inside" in the question was that it meant two different things in ONE use of the word.
Your comparison with "present" gives three different meanings in one sentence BUT only by utilizing THREE different uses of the word.
You did NOT write "present" once and it had all 3 meanings, you had to write it three times to get those three meanings.
Also the way you asked the questions to the AI already shows that you adjusted the original question to be easier to understand for the AI.
You asked the AIs specifically separated each time - one question only for a circle and one question only for a triangle.
If the original question would have been separated into 4 different questions (one for each shape) then the thought process would have most likely been different for the students than when it was put together in ONE question for all four shapes - especially with the circle being mentioned in the middle of the different shapes.
To use the AIs as a comparison you would have had to ask them the question like this:
"How many degrees respectively are inside a triangle, a rhombus, a circle and a pentagon?"
And that could have then tripped up the AIs trying to use the same meaning of "inside" for all four shapes, too.
I tried asking Chatgpt and Gemini that question and ChatGPT answered "correct" while Gemini answered that circles don't have degrees since it's not a polygon
It still would not have tripped up the ai, but thats not because it understood the question. AI is basically a very complicated and huge prediction machine that predicts what should be the next word, based on how many times it appeared previously.
So thus it would still connect the 2 questions separetly and give the right ans
As far as I'm concerned it's not really that the question has multiple interpretations but rather that it's just altogether malformed, like asking "how many meters are there in a kilogram?" - it's just gibberish. What does it even mean for a degree to be inside of a shape?
You are correct! I typed the question in exactly as written into ChatGPT and got this:
Triangle: The sum of angles in a triangle is always 180 degrees.
Rhombus: Each angle in a rhombus measures 90 degrees.
Circle: A circle doesn't have angles; it's composed of curves.
Pentagon: The sum of angles in a pentagon is 540 degrees.
I completely agree. Both AIs recognized the double entendre, hence why they had to clarify the question before answering.
You might complain that the question was poorly worded. But the real problem here was that the grader didn't understand the issue and marked the student's answer wrong rather than realizing that the student had answered the question as she understood the meaning of the word inside angle.
but the question never said inside angle :v
@@MAML_ said degrees inside, there are no university degrees inside a polygon so all answers are 0
At least the grader's quality is coherent with that of the textbooks and other teaching materials. "Enshitification" everywhere. 😠
The grader DID understand the issue. If you answer the wrong question correctly, it still is the wrong answer. If an ATM asks you to input your PIN, and you input your phone's correct PIN (which hopefully is different to your bank cards), you wouldn't argue that your answer is technically correct when the ATM refuses to tell out cash, would you?
This is such an insightful remark. What is the purpose of grading and giving marks? Are you just checking boxes or are you trying to understand if the pupil is willing and capable of thinking critically?
English is not mathematics; the homework question is malformed -- the phrase "degrees inside" is mathematically undefined and itself wrong to use in a mathematical context.
Yes, that is exactly the long and short of it.
So answer "360" means, that angles in absolutely all shapes are 360°. You can't equal 2 meanings: angle of arc and interior angle
This is correct. And easily demonstrated. I'd appeal the grade for the paper on principle.
This should absolutely be the correct answer to the question "as written".
Objection: Only closed, 2D shapes.
@@professorhaystacks6606 - true, but all the shapes in question match that description
@@professorhaystacks6606 The definition of shape is literally a closed 2D object. There are no non-closed shapes because it's in the definition that they're closed lmao
Bro is having fun with his editing software. Those glowing effects go crazy 2:18.
Fr😂
I hope that's not Premiere he'll have to learn something new when the TOS changes come lol.
@@donwald3436after effects
Me busting a move at 2:46
I think math questions demanding a precise answer should be asked precisely. Garbage in, garbage out, right?
Exactly! Math is all about precision.
So the Daughter was marked down for either having too much knowledge or sadly lacking in psychic ability or the marker assumed she didn't know that a circle has 360 degrees despite knowing all the correct answers for every other question. In conclusion, the marker is either a robot or doesn't have the necessary intelligence to accurately mark mathematics exams.
autism is one hell of a drug
or the marker knows his students and knows that we will get the best grade at the end of the year anyways and had some fun.
Or for not learning textbook material by heart. Common thing in schools tbh
Definition
AI : Artificial incompetence. 🤪
Years ago, I took the Praxis exam. I quickly realized you could actually know too much and actually get a bad score. I realized that for many questions none of the answers were correct. But that one answer was what a teacher would give in a classroom if they didn't really understand the subject. By answering the questions that way, I got a max score on the test. Which means it wasn't a very valid test. 🤣
If a question has multiple meanings it's only fair to accept multiple answers.
... but they should be limited to multiple choices.
By the logic the circle has 360° - every other shape has also 360°
why?
every shape does not have radius
@@syther836Because that's actually the sums of the compliments (edit: supplements, not compliments) of the interior angles. It's 360 for all shapes.
@@9adam4 Nope, Sum of interior angles of an n sided shape is 180(n-2)
@sytherplayz Read what I wrote again.
The reason the sum of interior angles is what you described, is because the sum of the COMPLIMENTS of the interior angles is 360 degrees.
I was once asked :- "Have you ever driven your car one handed?"
My reply was :- "No! I've had two hands since birth". 🖐😎👍
It was a glib answer but unarguably a correct one, even though it wasn't the information they wanted.
If a question is ambiguous then the any valid answer should be accepted as correct.
By saying “no” you’ve already given them the information they requested, haven’t you? You just also gave them extra information with the follow-up sentence.
@@divisix024 He didn't say he'd never driven a car _using_ only one hand (which was presumably the intent of the question), only that he'd never driven a car while _having_ only one hand. He certainly could have driven a car using only one hand while still having two.
@@divisix024 Not really! I often drive with only one hand on the wheel when changing gear but that again is not the info they were after. What they wanted to know, I would never admit to. Would I?
This reminds me of the story where two astronauts were on the moon and one bet the other that he couldn’t make donuts there. The other one jumped in the lunar rover and did donuts in the regolith. The other one said that THAT was not what he meant, the other replied that since the 1st one wasn’t specific, he (the second one) got to choose the definition of ‘donut’.
I generally agree that nonspecific questions should be accepting of any answer that technically fits, but the problem with people throwing homework questions online is that we are seeing them with zero context. For a student that has had a teacher telling them all week that circles have 360 degrees but, upon being asked, answers that a circle has infinite degrees deserves to have their answer marked incorrect. Maybe give them some kudos for a clever response, but they deserve to be marked off for not paying attention.
I’m with the daughter. The question was ambiguous. Mathematics is a science of precision, and so should its language be. Many years ago when I was in secondary school I was given a very challenging problem to solve involving the volume of a segment of a cone. The teacher had lifted the question from the national school leaving certificate exam but had changed one word, completely altering the meaning. She marked the test based on the original question and failed me because I answered the altered question. I eventually consulted with an expert mathematician who agreed with my answer. I never liked that teacher after that incident and I have never forgotten the incident over 50 years later.
Or, perhaps the class had just been taught about the '(n-2)*180' formula and the test was simply to find out if the kids could apply it correctly? Apparently, the mathematician's daughter could not.
@@undercoveragent9889 that formula is not applicable to the circle.
@@undercoveragent9889 Using the (n-2)*180 formula, the mathematician's daughter is _more_ correct than the """correct""" answer.
@@undercoveragent9889 Applying the formula in reverse to the desired answer, you discover that the circle has four edges. Are you sure this is the correct explanation?
@@undercoveragent9889 blindly applying a formula without thinking about what it means isn’t a habit you want to encourage children to do.
I think this would have been a great question to have asked _during a math class,_ so that students could discuss the ambiguity and gain deeper understanding. Tests shouldn't be ambiguous. They're for testing what you've already learned, not posing new questions you haven't learned about.
Because the question can be interpreted both ways, both 360 and infinite should be acceptable answers.
Only if the responder specifies their assumptions.
autism is one hell of a drug
@@crinolynneendymion8755 , well the questioner didn't
And by the logic of the circle, 360 should be the correct answer for ALL closed shapes! Either the question refers to degrees inside meaning sum of interior angles, or the absolute arc's angle, which always would be 360. And it isn't how presh explained with the present example, because it is using one generic instruction for 4 sub questions, so it would put the SAME meaning in all 4
If it doesn't then that is like a variable having 2 values you randomly take
This proving that all shapes are squares. 😅
My immediate thought on hearing the question was ‘infinity.”
180(∞ - 2) = ∞ - 360
My first thought was "There are no angles in a circle."
@@TonyCAV8R The answer is right but the solution and the notation are wrong. Infinity is a limit. It can be a result but (almost) never a part of an equation. Especially at this level.
@@Golfnut_2099 , I agree. By the definition of a circle, there are no angles.
Mine was zero, because I reckoned there are no angles in a circle.
I would have said 0. There are no corners in a circle, so how can there be any angles?
One could argue that there are an infinite amount of corners in a circle.
A circle is what defines angles. You don't need any corners. If you turn 360 degrees around, you will be right back to where you started, so there are 360 degrees inside any 2d object.
@@EaglePicking but then it wouldn’t be a circle and instead be a shape with a lot of sides
@@EaglePicking You could also argue there are an infinite number of angles inside a triangle and all but three of them are 180 degrees (the rest are located somewhere on the sides).
Once you have decided you are not going to do that, and instead you are only going to sum the angles located at a vertex (as we did for triangle, rhombus, and pentagon) , then since a circle does not have any vertices it does make sense that the answer for the circle is zero. At least that is one reasonable interpretation of the question.
@@nigelkearney5557 Yes, that is also reasonable.
I don't know about the first three, but there's about 68 to 72 degrees inside the pentagon depending on the season.
You mean « THE » Pentagon!
@@richardgratton7557 r/whoooosh
True, although in science and math homework, it would befit the teacher to ask for the answer in Kelvin instead. Which BTW should be around 294 K.
@@unvergebeneid You're correct! My bad...
Even then you have to be specific. When I read that, I read degrees Celcius, and they will all be cooked.
The stretched explanation for why it's okay to mix the use of a term in a math question despite not providing clear context is not great, tbh.
He didn't say it's ok. He says: "You would say you're using one word to mean two different things which should never happen in mathematical homework. This is a poorly phrased question. It should never be allowed." and I don't think the "different perspective" that follows contradicts this; it simply explains that it is a fact of every day life that you might come across ambiguity like this.
We are getting the question without context, but the student is not. The teacher undoubtedly taught what the answer was supposed to be in the special case of a circle. So the student rightfully loses those points for not paying attention.
Those were always the kind of question I were afraid of in math class because I know the teacher wasn't looking for the correct answer but for answer they thought was correct.
That was their fault, not yours.
But it'll be your grade, not your teachers.
@@minor_2nd Ngl, this make me respect the curiculum in my country more. At the end of each test, student is allowed to recieve back their test paper and if there exist ambiguous questuon such as these, we are allowed to discuus it with the teachers. And we are also given extra make if the question is indeed ambiguous as a bonus question mark :D.
Heh, I still remember having boxes labelled 1st-7th and we had to write in the days of the week. I got the order correct, but started on Monday instead of Sunday, so each one was marked wrong. I maintain that the week starts on Monday. Justification: we call a Saturday and the following Sunday "the weekend", singular.
@hughcaldwell1034 but Sunday is the day of Sun, which is 1. Moon is 2, Mars is 3, Mercury is 4, Jupiter is 5, Venus is 6, Saturn is 7.
My immediate thought was "wouldn't the answer be 360° for all of them?"
I think my instinct was to find the interpretation of the question that best fit all of the shapes, to which the inclusion of circle in the list limited most strongly.
I don't imagine that would have gone over any better in the grading...
You'd have got 50% or 360% on this test depending how the teacher felt that day
Having a Field’s medalist’s daughter as your maths student must be terrifying
Evidently not terrifying enough.
@@unvergebeneidyep, cause that teacher didn’t do his/her homework and mark the kid wrongly.
@@pnoodl3s775 exactly. And also because they gave their class this terrible homework assignment in the first place.
There should be no ambiguity in maths tests like that. Either the question as written should have been thrown out, or they should have accepted both 360 and infinity as answers. Even “undefined” should have been acceptable because relative to the other shapes, a circle is not a polygon so cannot have a sum of interior angles.
lol How on earth do you make (0-2)*180= infinity? The fact is, in order to construct a circle from an infinite number of pairs of intersecting lines, you would have to draw an infinite number of vertices... which is impossible. But guess what: we _can_ construct circles using just a compass. The circles _you_ imagine cannot even exist because there are precisely _zero_ straight lines along _any_ segment of any circumference of _any_ circle. By definition, the circumference is equidistant from the centre at every point but if there is _any_ straight line along the circumference, different points along that line will be at different distances relative to the centre.
In any case; I have _never_ inscribed angles of infinity degrees when constructing any circle.
There is a lot of intellectual snobbery in this comments section and it is purely for the purpose of justifying a mathematician's daughter who failed to apply (n-2)*180 correctly. Why are you all simping?
@@undercoveragent9889 hahaha! Not sure what’s worse, this snobbery you refer to, or massively overthinking things and coming out of the other end looking like a fool. Either way, appreciate the entertainment!
Arguably, a shape that is not a polygon can have a sum of interior angles. For example, a quarter disk (or, bluntly speaking, a slice from a pizza that has been cut into four equal slices) has three interior angles of each 90 degrees, and hence its sum of interior angles is 270 degrees. So I'd say that the sum of interior angles of a circle (or disk) does exist, and it is 0 degrees.
By the way, not sure what the other replier is on about, but the (n-2)*180 formula only applies to polygons, i.e. shapes with straight (= non-curved) edges. It does not apply to shapes with curved edges (such as, for example, the quarter disk and the circle).
@@undercoveragent9889 It is not 360 degrees either, and that formula is not valid for any non-polygons (like a circle).
But it just so happens that the circle is the limit of the regular n-gons as n goes to infinity, making 'infinity' (while a not strictly correct answer) a better answer than '360'
@@undercoveragent9889Even if practically impossible infinite vertices are mathematically possible. You also cant even draw a perfect square in real live should that invalidate the formula?
I don't think the question is fair. the purpose of a test/home work should never be to trick the test taker. The goal of a test/home work is to challenge the student while ensuring they learned the concept.
When I asked chatgpt this question, here is what it said:
Conclusion:
The test is not entirely fair due to the inclusion of the circle. A circle does not fit the same criteria as the other shapes when discussing interior angles, which could confuse students. To improve fairness, the circle should be replaced with a polygon, such as a square or hexagon, or the question should be clarified to avoid ambiguity.
Unless, of course, the teacher spent all day telling the students exactly what the answer was for a special case like a circle. When people post homework questions online, we are viewing it entirely out of context. That student that was in the class was probably told how they were expected to do it and decided to not pay attention, thus, they get points marked off.
lmao, I love how people use LLMs to justify their reasoning, like they can never be wrong and aren't tainted by popular beliefs / misconceptions
@@SgtSupaman And this is how we end up with Galileo Galilei under house arrest for the remainder of his life. Church told him the universe is geocentric, clearly he wasn't paying attention.
@@Eidako , wow, thanks for that wonderful example of an Appeal to Extremes fallacy. You do know a circle actually is 360 degrees, right?
Don't get me wrong, a great teacher should absolutely give some kind of credit to a student that comes up with a clever response, but if they were taught the proper answer and can't supply it when asked, they aren't paying the attention they should be paying, which is an appropriate lesson in and of itself.
@@SgtSupaman There is no proper answer to this question - there are three quite valid responses to it. A circle is a 360 degree rotation, yes. However, a circle is not a polygon, meaning in the context of how many degrees are in a triangle, rhombus, etc., it has zero vertices and therefore there are zero (or "not applicable") angular degrees inside. Alternatively a circle can be considered to be a polygon with infinite vertices (the 3D modeller's approximation), in which case there are infinite angular degrees. The girl was rightfully confused by it.
I wonder how the teacher would describe the degrees inside a shape that is half of a hexagon on one side and half a circle on the other side?
The logic applied to the circle should be the same for the other shapes. If the circle has 360 degrees then they should all be 360 degrees.
I was wondering about 0 being correct as there are no discernible corners in a circle.
Another perspective:
For polygons, we're looking at the sum of all interior non-straight angles. If we counted 180 degree angles, then all shapes would be infinite.
As n approaches infinity for a regular n-gon, all interior angles approach 180 degrees.
Would all interior angles for a circle be 180 degrees? If so, we exclude all of them, and get an answer of 0 degrees.
As far as the fairness of the question goes, I'd rate it as not fair. It's an unrelated fact memorization problem tossed in the middle of measurement problems. It's also obnoxious that the circle breaks the sequence of shapes, 3-4-infinite-5.
In instances such as this, all reasonably justifiable answers should be accepted. 360, infinity, 0, undefined.
Another: There is one revolution in one revolution. The number of steps you divide it into is arbitrary.
The presents example is not the same as the homework question as each instance of the word present had a specific meaning and repeating the same word is fine, but in the homework question the word inside is used only once and the way it is meant to be interpreted inside has multiple meanings which is confusing and not standard mathematical practice.
I think its a fair question, but if the kid can justify infinity to their teacher, it should *absolutely* be considered a correct answer. 360 or infinity both make perfect sense, depending on the interpretation of the question.
How does infinity make sense? It doesn't and neither do you.
There was room to give both answers and a short justification.
The problem is that some teachers doesn't accept alternative answers.
It happened to me in a class of the first semesters in Engineering.
@@luismigueluribe914 Depends on the explanation of the alternative.
@@undercoveragent9889did you watch the video?
Infinity is correct IMHO. Take this assignment back to teacher and ask them "How many degrees are in a thermometer"
"Inside" a thermometer! 😅
@@RyguyAB How many degrees are inside 1) triangle 2) circle 3) thermometer 4) (x-3)(x²+7)(x+4)
It's like being asked the question, "How much dirt is in a hole 2' wide, 5' long and 4' deep?"
Because the question was ambiguous, the test taker should score both answers as correct
Question is not fair because the structuring implied that the same question is being asked of all 4 shapes when in fact the circle question is different. Puttin the circle question in the middle of the other shapes sure makes it seem like you're asking the exact same question 4 times and not 2 different questions.
As a high school Geometry teacher, I would have given credit for the answer of infinity but used it to then have a discussion with the class so they understood what answer I was looking for.
This!
i probably would have nocked down infinity. maybe partial credit. i would not have accepted 360. full credit would have gone to not defined. given it is elementary i could see a strong argument for zero as circles don't have interior angles therefore the sum of nothing should be nothing but it is in the end not correct. probably would give it half credit.
this question is nonsensical; you should not give tests with questions like this. ask what is the sum of angles of the following figures
The best mathematicians-to-be in the class will say infinite. The best test takers (often times the top of the class) will say 360. The best logisticians in the class will say both.
You're right... it's a poorly constructed question.
Deep down, I'm compelled to say infinite, because you can prove it logically.
what about not defined
The real problem is that an answer without a reason was expected. The reasoning behind an answer, in school, should hold more value because you are trying to get students to actually think. A far better test would ask for ‘why?’, in which case both answers would have been acceptable. A “correct” answer with the reasoning being “Because that’s what I memorized” should only get partial credit.
Or, perhaps the class had just been taught about the '(n-2)*180' formula and the test was simply to find out if the kids could apply it correctly? Apparently, the mathematician's daughter could not.
@@undercoveragent9889She did, if you take n=infinity for the circle.
@@undercoveragent9889 The (n-2)*180 formula only applies to polygons, which are shapes with straight edges. A circle is not a polygon.
That being said, the daughter did give an answer that corresponds to the (n-2)*180 formula if a circle is considered a regular "infinite-gon".
@@undercoveragent9889 She did with the other shapes. It only works for polygons.
@@undercoveragent9889 You appear to be arguing that a circle has four sides: if (n-2)*180 = 360, then n=4
If that is the purpose of the question, then it's not just a bad question; it's a terrible one.
3:04 You’re just having fun with the audio and visuals today, huh😂
Assuming _internal_ angles are meant, it's 11) 180°, 12) 360°, 13) N/A, 14) 540°.
A polygon with n = ∞ isn't a circle, it's a polygon with an infinite number of sides. Circles don't have sides, so they don't have internal angles. "Inside" would have to mean different things depending on the shape for a circle to have 360°.
THIS is the correct answer
The problem is that a degree isn't a thing on its own. It's a unit of measurement.
An equivalent question would be asking how many centimeters are inside a shape but for some shapes asking about the circumference while for the circle asking about the diameter.
Infinity is correct within the context of the other shapes. If the teacher forgot about this, both answers should be correct. If the teacher deliberately meant it as a trick question, he shouldn't be a teacher anymore.
These are the type of questions where you have to pray that your answer is not correct but what the teacher thinks is correct .
Why don't you Google it: 'how many degrees in a circle'. Make sure that you are sitting down when you do. The shock might be too much for you.
I stopped worrying about what my teachers BELIEVED in 3rd grade. Math and science were presented at such a low level that it wasn't until college that I needed to "study" them at all.
The existence of homophones and the answers given by LLMs are irrelevant; maths is supposed to be unambiguous.
Maths might be, but the English we use to discuss it definitely isn't. Linguistic ambiguity generated by translation is common.
Science has specific definitions to avoid ambiguity. "Velocity" in science is a speed and direction, a vector. In English, it can be the same as speed. "Speed" is a scalar in science with no direction. It is not a vector.
Let me guess, you're at best an undergrad?
Ah yes, the famously unambiguous maths. Where sin^2(x) and sin^-1(x) mean completely different things, where A’ can be a transformation or a derivative, and where the unwritten logarithmic base is always 10 except when it’s 2.
@@fieuline2536 except except when its base e
one of the mistakes I made in the past was about ambiguous questions, like the relative position of two lines, I used the question 'what can we say about the two lines (d1) and (d2), until one of the students wrote: 'they are beautiful' , the answer stunned me and I thought: 'this can be an answer if he sees them like that' and I gave him the full mark and never repeated an ambiguous question again.
I’m a math teacher. As long as the student could defend infinity I’d accept it.
Meanwhile, what is the central angle of all plane, convex polygons?
Planar polygons are not real objects; they're mathematical constructs. The answer depends upon the arbitrary number that you choose to represent one revolution in your system. 360 was chosen by some recent inhabitants of Earth, others chose two (2*pi) or 100. It could equally well be one (1). Martians could have a system based on their ~670 local day orbit.
@@psdaengr6155 I'm aware of the choices 2pi, 360 and 400. Who did use 100?
@@__christopher__ 2*pi has never been used for degrees. it is used for radians. i'd argue that the way degrees are traditionally defined the polygons would have degrees but a circle in fact would have to be considering the central angle which is more properly defined in radians then degrees.
I have another fun method to solve for the sum of interior angles. Since, the sum of interior angles can be given by 180°(n - 2), where n is the number of sides. Instead of taking n as infinity we can also take n = 0 for a circle. Now when you subsititute the value in the equation you get the sum of interior angles in a circle to be -360°. It seems absurd at first glance but if you think about it, the negative sign actually refers to the measure of central angle instead of sum of interior angles. We can also play around with the values of n to verify the logic of negative sign being the measure of central angle. For example if n = 1, which would refer to a straight line, where the equation gives the sum of interior angles to be -180° or measure of central angle to be 180°. Again if n=2, then the structure will be for any two lines joined together at a point, where the sum of interior angles is 0°. Which for me it again makes intuitive sense because the structure is not well defined. So even if we do not know that the measure of central angle in a circle is 360°, we can conclude the same using the equation.
Mathematically speaking I would argue that the question was not poorly phrased because if it were then the equation should not have come to the same conclusion.
true, as you said that a word can have multiple meanings in sentence. but as you said right before it a word in a mathematical question should *never* have multiple meanings
2:46 Griffpatch outro music goes hard 🔥
Circles don't have corners, so there is no definable answer.
That itself would have been an answer. This wasn't multiple choice, and wasn't limited to a number.
Math circles don't actually exist.
@@psdaengr6155 X^2 + Y^2 = 5 is imaginary?
Also, if the pentagon is concave, the answer gets more complicated.
@@jwill7998 Would depend on the radius of the concavity.
Your graphics are top notch, and track seamlessly with your narrative - begging the question “Are your narratives top notch?” Well duh, yes your narratives are top notch also. Thank you for your time, effort and passion. You are endlessly entertaining and thought provoking.
I was briefly a math teach after leaving the U.S. Navy (as a civil engineering officer), I taught high school algebra and geometry. Whilst going through high school (like most of my peers) I disliked word problems. Teaching algebra and geometry led me to the realization that word problems are the superior method of imparting wisdom and analytical thinking.
Your videos are a wonderful expression of both wisdom and analytical thinking and if i were still teaching HS subjects I would be discussing algebra and geometry using your videos as much as possible.
Bless you and Bravo Zulu on your efforts and achievements.
If there are 360 degrees in a circle, then there are 360 degrees in all those other shapes as well, as we defined "degrees in a shape" to be "degrees around the center" rather than "degrees of the angles between the sides"
Though once you get hexagons and higher, you get shapes where there is no point which can "see" the entire interior of the shape, which makes "center" a murkier concept.
I suppose you could phrase the concept in terms of maximum winding number...
You could argue a circle is 360° because a circle is made of two half circles, which each have 2 angles of about 90 degrees. So 90° x 2(amount of angles per half circle) x 2(amount of half circles per circle) = 360°
Mathematicians, unfortunately, do not write the math questions for elementary school homework.
You don't even need to be an actual mathematician to write competent questions. Just a basic understanding of what you are asking will do.
*I don't agree with that perspective: **9:04**.*
We don't have a "single word with 3 different meanings", we have "3 different words that are written the same", at least that's how I see it. For your example to work, you had to write the word "present" 3 times, that didn't occur in the polygons question, the word "inside" was written just one time.
So, for me, if I have one question to be answered, we should apply the same logic to all of its items
There are no straight lines inside a circle. The sum of the angles could be interpreted as 0 or infinity. That would mean it is undefined. If you’re looking for the central angle, then yes, 360 degrees. Then you could logically argue that all the shapes could have a central angle of 360 degrees.
this was a rollercoaster of a video, i was initially annoyed at the teacher, but then your explanation about the English language and how the meaning can be inferred, that was beautiful.
Is it weird, that i felt moved with how nice this video is. purely logical explanation and by the end of it there is no finger pointing.
I would have written "undefined. No angles." I think the question implies asking about interior angles.
The empty sum is well defined and zero.
Poor phrasing in poetry prevents getting to the meaning.
Poor phrasing in math prevents getting to the moon.
If the person asking the question knows that there are multiple ways to interpret an angle being "inside" a shape, then the question should have specified which kind of measurement was intended. If the desired answer involves using multiple distinct interpretations of "inside", then clearly the author has multiple interpretations in mind, and therefore the question was deliberately ambiguous, and a deliberately ambiguous question does not have just one right answer.
Before watching this video I recall that the sum of the (inside) corner angles of a triangle is 180°. like a 60-60-60 triangle with identical angles and side lengths. For every (inside) corner you add to the polygon I think you add a fixed number of degrees. Maybe 180° for every extra corner. Because a square has 4 x 90° = 360° (180° more than a triangle. Thus:
11) Triangle = 180°
12) Rhombus = 4 corners = 180° + 180° = 360°
13) Circle = zero or ∞ corners (maybe 0° or ∞°). My answer is Ø (null)
14) Pentagon = 5 corners = 180°+ 2(180°) = 540°
EDIT: After watching most of the video I agree that "how many degrees inside", as stated in the original question should be interpreted as "the sum of interior angles". As there are no actual angles or corners along the line of a circle, 0 may be considered the most "correct" answer.
Channeling a bit of Grant Sanderson there in the animation. All that’s missing is the pi icon.
Fun fact: How to tell if a point is inside a closed figure, and how to orient handedness of the figure (in 3D)
Choose any point P on the plane.
Choose any point S on the figure.
Define a vector V = P - S.
Define A = arg(V) => the angle of V wrt any origin.
Integrate A over a closed loop path of S once around the figure.
If A is outside the figure, Int(A) = 0.
If A is inside the figure, Int(A) = +/- 360 degrees.
The sign of Int(A) depends on the direction of the path integral and so gives a handedness orientation
By convention, right handed equals a positive integral for a counter-clockwise traversal (from x-axis towards y-axis yields positive z-axis).
Since the choice of origin was arbitrary but does not affect the result, the result is invariant wrt translation and rotation.
I think the sides of a circle do not have angles. Therefore the answer is 360 or none.
or not defined and what would be the difference between 360 degrees and 720 degrees or 1080 degrees?
The other way to avoid ambiguity is to draw in straight lines from the vertices of each polygon to its geometrical centre in a radial type way. The angles, therefore, at the centre point would always be 360° in any shape or, in other words, the central angle will always be a circle around the geometrical central point, - so that could refer to “angles total inside a shape“ in all instances.
Other than that, the ambiguity raised in the video was a good one. The girl’s response of infinity for the circle was a valid one based on the reasoning given and it should’ve been clearly stated what they were referring to when asking the question! A very good video and very informative. Excellent.🌞
My immediate thought was that the question was meaningless, and one needed to re-write the question in a meaningful way before answering it. It is not clear why the vague and almost meaningless phrase "how many degrees inside" should be translated as "what is the sum of the interior angles expressed in degrees of."
No, that wasn't the question. Clearly, this was a geometry class for kids where they were being taught how shapes can be constructed. The formula '(n-2)*180' computes the sum of all the angles that have to be inscribed in order to construct shapes where 'n' is equal to the number of pairs of intersecting lines or, 'the number of vertices'.
Or, keeping it related to trig: the question refers to the minimum number of triangles are required to create regular polygons. It takes one triangle to make a triangle, two to make a square, three to make a pentagon, ... and each triangle adds another 180° to the sum of all angles that must be inscribed in order to construct the shape. You cannot construct a triangle without inscribing three angles that sum to 180; you can't construct a square without inscribing angles that sum to 360 and you cannot construct a circle without inscribing an angle of 360°.
The simple application of (n-2)*180 holds in every case unless of course you are the daughter of a mathematician when '(0-2)*180' somehow equals infinity.
@@undercoveragent9889 how can you correctly state that this formula by definition is applicable only to regular polygons and also fail to see that circle is not a regular polygon, making the formula not applicable?
@@undercoveragent9889This is so true. A circle has 4 sides as given by the formula. We should question nothing and believe whatever goes in our ears🎉🎉🎉
@@undercoveragent9889 the formula is very specifically defined for polygons only and requires significant constraints to work for concave polygons at that. it is not defined for non polynomials such as any shape with a curved edge like a circle.
@@somewhatfunnyguyydon't ask questions just consume the product and then gets excited for next product😂😂😂😂
There are no internal angles in a circle, and this question implies internal angles, so the answer is N/A. I would also score 0, 360 or infinity as “correct”.
0:41 the formatting is so bad
Depending on the grade of the student and with the question being an "easy" one here (with the triangle, rhombus, and pentagon), then it is obvious the answer the teacher was looking for is 360 degrees.
The student can most likely bring up the reasoning and ask the teacher for the point back, but if this was a real final exam, then there would have been given a point for infinity, as student do sometimes see answers that the question makers dont.
One example I have seen is (do note the question isn't originally in English):
You have a roulette wheel with 2 labels of "spin again", 1 label of "10% off", 1 label of "20% off" and 2 labels of "nothing", to determine the amount off you get from your purchase.
What is the chance you get "nothing"?
What is the chance you get "20% off" ?
The answer key said: 2/6 or 1/3 for the "nothing" label, while 1/6 for the "20% off".
But some student submitted answers of "1/2" or "2/4" for the nothing, and 1/4 for the "20% off".
I can see the official gradting notes does indeed state that those student answers were accepted as correct even though it didn't match the official grading key.
The only correct answer is "What do you mean 'degrees inside'?"
My teachers hated me for that, but indeed that's always the right answer when the question is poorly phrased.
EDIT: also, what if there was a circle or a square inside the triangle? What would that mean? How many angles would it have inside? How does phrasing the question like that helps kids learn when they haven't developed this kind of subtletly in their thinking yet (or even teenagers)? Except if the teacher has got a plan to make them think through it and then make a point out of it later, rather than simply grading it as "right or wrong", that's really harming more than helping.
I would say the daughter is more educated than the teacher... she took the meaning of inside saw that majority fit this definition and used that definition to come up with the answer for circle... that's how math started... you establish a rule and then solve stuff according to the given rule...
Or, you could break it down by cases, and answer, "If by 'degrees inside a triangle' you mean the sum of the triangle's interior angles, the answer is 180. If, however, you mean the sum of the angles subtended by the sides of the triangle with respect to any point in the interior of the triangle taken as a vertex, then then answer is 360," and continue in that vein.
@RolandHutchinson It is a good option and is usually what I do as an adult in conversation (not usually when answering written questions on exams because I expect those to be properly phrased by default, unless the teacher is willing to make a point out of it and make students think, but they usually don't, in my experience. I don't know what this particular teacher was going for, so I can't speak about this particular case).
I don't think middle schoolers or even high schoolers usually have that level of mental flexibility or communication skills, though, even the smarter ones, especially because that's not usually how subjects are taught at school. Answers are either this or that most of the time and most teachers don't welcome lateral thinking and exploration very well (again, not the ones I was taught by, even in college). I certainly wouldn't have come up with that answer at that age, and I was thinking more on that vein, and I don't think unclear (exam/homework) questions deserve elaborate answers either way.
"Real", adult life is a different thing. If you're trying to sort things out with someone on your job or personal life, it is not necessary to be nitpicky, and it is better to elaborate and make sure everyone is on the same page.
@@MultiSenhor Nicely put!
I would just write "Not enough information to answer question properly."
According to the teacher, a circle and rhombus both have 4 sides.
The inside the outside the upside and the downside.
@@peterdavis9403 Hahaha
@@peterdavis9403 right side and left side plus observe side and reverse side.
I posed this question to ChatGPT -
Give the number of degrees inside each of the following figures - (a) triangle, (b) rhombus, (c) circle, and (d) pentagon.
Its answers were -
Sure, here are the number of degrees inside each of the figures:
(a) Triangle: The sum of the interior angles of a triangle is always 180 degrees.
(b) Rhombus: A rhombus has four interior angles, each of which measures 90 degrees. So, the total number of degrees inside a rhombus is 360 degrees.
(c) Circle: A circle does not have interior angles in the same way polygons do. Instead, it's measured in terms of its central angle. A full circle has 360 degrees.
(d) Pentagon: The sum of the interior angles of a pentagon is given by the formula (n-2) * 180, where n is the number of sides. For a pentagon (5 sides), the sum of interior angles is (5-2) * 180 = 3 * 180 = 540 degrees.
My first thought was "There are no angles in a circle."
The answer should be ZERO!
Then define 'quadrant'. If there are no degrees in a full circle then the number of degrees in a quadrant would be 0/4, wouldn't it? Maybe you have misunderstood something very fundamental?
@@undercoveragent9889 Circle: a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center).
Quadrant: each of four quarters of a circle.
We are talking about different things here... The degrees of the inside angles of a polynomial are not the same as degrees of arc of a circle.
@@Golfnut_2099 "Circle: a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center)."
Yes and further, the circumference inscribes an angle of 360° around its centre.
Given the context of the geometry lesson being taught to these kids and that they were being taught how to construct shapes accurately using only a pencil, a compass, a protractor and a ruler, this 'test question' appears to be aimed at testing how well the students apply the formula (n-2)*180. In essence, all of those defending the mathematician's daughter are suggesting that the formula does not hold for 'n=0'.
Maybe the question should have read:
"What is the sum of all angles that must be inscribed in order to construct the following shapes?"
But, I think that from these kids' point of view, that would simply be a paraphrasing of:
"How many degrees in the following shapes?"
The fact is: you cannot construct a circle without inscribing an angle of 360°. Sadly, Presh missed an interesting implication of how circles and triangles are related. In fact, you _could_ imagine the centre of the circle as being the 'apex' of a flat cone and if you cut the cone down its centre, you get a triangle. Right? You _could_ think of a circle as being a line that is parallel to a line through its centre and perpendicular to its plane and what you have is _two_ lines that do not intersect.
Quadrant, a 90° section of a circle. What about 'sextant'? Any relation to 60° at all or are we going to continue along the disingenuous path? If there are 60° in a sextant, 90° in a quadrant then in what way is there _not_ 360° in a full circle?
@@undercoveragent9889 Now divide a triangle into "quadrants" around it's centre. How many degrees in each quadrant?
@@undercoveragent9889 but if that is how we are defining "degrees in a" then we should be looking at the central angle from the center of each shape to be consistent and every shape listed would have 360 degrees inside it.
Questions like this should be used to test kids creativity in coming up with an answer
my answer would be zero, because the inside wall of a circle is a curve, so there are no points of intersection of straight lines to measure interior angles.
Points, lines and circles are all imaginary objects.
I probably would’ve given both answers in that situation. I’ve been doing that and leaving a little note next to the question my whole life explaining the ambiguity of the question.
Does that mean a circle equals 2 lines?
That is exactly why a line is not 180 degrees. 180 degrees is what a line segment does when it terminates and returns to its origin.
@@phillipsusi1791 What you describe is an _exterior_ angle of 180 degrees; the adjacent interior angle is then 0 degrees.
Yes, but only if you bend them just right and glue them together.
triangle has 160deg = half rotation
square has 360deg = full rotation
from there its easy to remember, that every additional corner adds +160deg, adds a half rotation -> anglesum = (cornerount-2)*160deg
circle is degenerate/asymptotic/undefined case, where infinite corners are 2+infinite half-rotations, which is nonsense. A more reasonable substitution is "no corners means; 0 MINUS 2 half rotations == -2 half rotations, to at least evade infinity, the sum of 0 corners in a triangle adds up to MINUS 360degrees. This is mostly fine in the way that spinors work, or how you just rotate in the opposite direction, or in how you can just abs() the result, ignoring the sign.
I think the question was fair, however, answers of 0, 360, or infinity should all have been acceptable correct answers.
and not defined as the idea of an interior angle does not exist for a circle.
His daughter should have raised her hand and asked for clarification. I always did this, and it "saved" me from reduced marks and later frustration. As a bonus, it's nice to see the faces of teachers when they realize they messed up, after you point out the inconsistency.
Why don't you Google it: 'how many degrees in a circle'. Make sure that you are sitting down when you do. The shock might be too much for you.
I would argue that the answer is 0, as a circle has no interior angles
imo all 0, 360 and infinity should be valid answers
The question did not say *interior* angles. Therefore, all 2d objects have 360 degrees "inside" them.
@@trollar8810 not defined should be acceptable as well.
I haven't watched the video, but the answer for any convex polygon would be n * 180 - 360 = (n - 2) * 180, where n is the number of sides. This is obtained by putting a point inside the polygon, and then drawing rays from that point to each of the vertices. You will end up with n triangles, each of which has 180 degrees inside it. We then need to subtract the angles where they all meet at the point inside the polygon, which add up to 360 degrees. So, for a triangle, we have 1 * 180 = 180, a quadrilateral would be 2 * 180 = 360, a pentagon would be 3 * 180 = 540, and a circle would be infinite.
Answer for a circle is 0.
The answer is the sum of interior angles. These points are corners in the shape. Since a circle has no corners, there are no angles to add up and thus the answer is zero.
Exactly. You must be an engineer like me 😉
I came to the same conclusion by measuring all the (non-existent) angles in a circle in my mind.
No, you can divide a circle into as many angles as you wish from the center. The sum of them will always be 360 degrees. Of course, this is true for ANY 2d object.
My friend, sadly gone from us, who was a games programmer in the 1980's decided it was more practical to have 256 degrees in a circle. He built sine and cos tables accordingly and it made his life easier becase of the 8-bit wrap.
Of course it’s infinity.
Why don't you Google it: 'how many degrees in a circle'. Make sure that you are sitting down when you do. The shock might be too much for you.
@@undercoveragent9889 You clearly don't understand the issue at hand and you seem to think the almighty Google is the world's superior problem solver. I suppose even the Fields medalist and his daughter fold at the hands of almighty Google.
@@undercoveragent9889 Here is a shocker for you. Google "How many stars are in the Big Dipper?" You will find 7. That is incorrect. The correct answer is 8, as Mizar and Alcor make up a double star. Google is not a reliable source of information. All that Google does is take your search query and find another source that it thinks is relevant. You can publish absolutely anything you want online and, if Google thinks it is the most relevant result for a search, Google will present that information first without any sort of verification.
Funnily enough, even searching "does google verify information" will not give you what you are looking for. It gives results for how Google verifies _identity_ for accounts, not information presented after a search.
"How many degrees inside the following shapes?"
360, for each of them.
Consider pizza slices from a pizza. If we sum the angles of the tips of each slice, the sum will be 360 degrees, regardless of the (convex) shape of the original pizza.
(Note: the word "inside" in the question implies that the tip vertices of all slices correspond to some arbitrary "central" reference point _within_ (= not _on_ ) the perimeter of the original pizza. However, if the tip point of the slices could have been chosen to be _on_ the perimeter of the pizza, then the sum of angles could have been 180 degrees instead of 360 degrees.)
However, if the question meant the sum of the inner angles at the _vertices_ of each shape, then the correct answers are:
11) triangle: 180
12) rhombus: 360
13) circle: 0 (and not "infinity"; a circle doesn't have any vertices)
14) pentagon: 540
Why don't you Google it: 'how many degrees in a circle'. Make sure that you are sitting down when you do. The shock might be too much for you.
@@undercoveragent9889 What exactly should shock me? That you can find a lot of mis-information when you Google things?
1) A circle is not a polygon, so "infinity" is wrong. Adding angles forever will still never create a circle. Think of it this way: The limit of the polygon approaches a circle, but what does the limit of one of the interior angles approach? 180°. So if you were able to reach the limit, you would simultaneously have a circle and a straight line. At the limit, the shape becomes undefined.
2) Degrees are degrees, they are not different meanings. An interior angle of a polygon is a measure of arc swept by those arms. Circles are 360 degrees of arc.
3) The question is not ambiguous. The primary lesson is about understanding the relationship between circles and polygons.
4) Most fundamentally, if a circle is anything other than 360°, then measuring angles becomes meaningless. The degrees of angles are what they are BECAUSE a circle is 360 of them.
The exterior angles of a circle can be argued to add up to 360 degrees, but not the interior ones. The question specifically mentions interior angles.
For interior angles, we can either argue that a circle is a limit as a polygon approaches infinite vertices, in which case, you have an infinite sum of numbers infinitesimally smaller than 180 degrees, or that the sum of interior angles is 0 degrees, since circles don't have any vertices (and note that for normal polygons, we're not counting any of the 180 degree angles that we get when we pick some random point somewhere along a side. Only vertices count!).
@@Keldor314 Your first two sentences are false. A circle has no angles, a circle is made of arc. A circle is 360° of arc, or the whole set of points in a plane equally distant to a given point. The question specifically says "degrees", not "interior angles". "Angles" is not in the problem at all.
A circle is not a polygon. If you can't grasp this fundamental idea you will never understand the problem. A near-infinite sided polygon is an approximation of a circle, useful for some mathematical approximations, but it is not a circle. If you say "circle" then you are not talking about an infinite sided polygon. These are two distinct things with different properties.
Circles have no angles. Circles have 360 degrees of arc. It is degrees of arc that you are measuring when you measure an angle inside a polygon. The number of degrees in that angle measurement is a direct reference to the 360 degrees of arc in a circle.
The whole point of this problem is to highlight the relationship between polygons and circles. Most of what I have said here is just a repeat of what I said above, you just need to read carefully and think about it. And read the original problem again.
@@TashiRogo Alright, but none of the polygons have any degrees of arc at all, not being constructed out of arcs, so clearly the question is conflating two poorly matched constructs. So should we add a refrigerator to the question so we can add the degrees Farenheit to the mix? Or maybe the one degree of freedom the refrigerator door hinge has?
The problem with mixing interior angles with circles in this way is that we've deftly replaced one measure by its counterpart. Imagine you take the vertices of a polygon and replace them with arcs, "rounding off the corners", so to speak, so that you're left with a smooth curve. How many degrees are inside of it? Well, 360, of course, since you add up the lengths of the arcs after normalizing them (and subtract ones curving "outward"). But this doesn't match the amount of degrees it had before unless it started as a quadrilateral! What happened? Well, those arcs correspond to the *exterior* angles of the polygon, not the interior ones. The fault lies with the choice of defining a straight line as 180 degrees and a sharp turnaround as 0. They're backwards.
So I guess I agree with you about the inside angle of a circle. It was the polygon examples that suffer from unnatural definitions of angles. I suppose we should measure angles as if we're bending a wire instead of unfolding a piece of paper. How did that folded over piece of paper get considered the starting point anyway?
@@Keldor314 They aren't poorly matched constructs! 😆 The whole point of the problem is to show how polygons and circles are related to each other. Circles have a whole set of properties that make them useful for solving various kinds of problems. Polygons have a different set of properties that can be used. Degrees can be applied to both, and by using this relationship, you can combine the various properties of each together to create a bigger tool-set.
It's easy to take for granted what it means when we label angles with some number of degrees. Lets take a square for example. It has four 90 degree angles, but what does that mean? 90 degrees of what? What it means is that if you drew a circle centered on a vertex of that square, 90° of that circle would be inside the square. It's the same for any angle of any polygon. The degrees of the angle is a measure of how much of the circle that is centered on the vertex of that angle would be inside the angle. So, although polygons contain no arc, and circles contain no angles, the angles in a polygon are measured in degrees of arc, and the degrees of arc in a circle are demarked using a central angle.
Also, the central angle only works for circles because they are completely uniform with respect to their central point. The central angle of a polygon is a nonsense concept. It doesn't measure anything useful. If you wanted to measure a polygon where you have rounded the corners somehow, you would have to break it down into smaller parts and use multiple measurement concepts to accomplish this.
Once you understand this, you can see how silly it is for all of these people to say that the student's answer of infinity degrees for a circle was a good answer, or how zero is an acceptable answer. The degrees in the circle are what is used to measure the degrees in the polygons. If you say the degrees of a circle is not 360, then saying that a triangle contains 180 degrees doesn't even mean anything.
@@TashiRogo All regular polygons have 360°, the same as a circle. That's how they're related.
and a spherical triangle has 270 degrees total interior ( 3 @ 90) start at the north pole, head south to the equator, turn left 90 degrees, travel 90 degrees along the equator, stop, turn left 90 degrees and head back north to the pole and you find yourself facing 90 degrees ( perpendicular) to your original path thus 3 points of 90 degrees each making it 3X90 = 270
The sum of supplementary angles of any shape is always 360°.
If you have a triangle that would give three angles a1, a2 and a3. For each angle, you'll get 180-a1+180-a2+180-a3=360 which gives a1+a2+a3=180.
You can generalize : for an x angles shape, you'll get sum (1 to x, 180-a(x)) =360
Which give sum(1 to x), a(x)) = 180 (x-2)
Also said, the sum of a x sides shape angles is 180*(x-2)
A circle can be considered as an infinite sides shape thus having a sum of interior angles of infinity.
I agree, as the question is stated 'Infinity' is the correct answer ...
..to any point on a circle from the center u can turn a maximum of 360 degrees to go to any poin on the surface, however thats not the only angles 'inside a circle',
we can do this two ways... by deviding it into n triangle slices aka n-gon, [180(n-2)] deg aproaces infinity as n increases...
of adding angles between points on the surface of the circle..and as ther is infinity many points thers also infinity many angles... the sum of angles goes to infinity...
the only matematically correct answer is Infinity...
..Physically u can argue the smallest distance between points on the edge would be The planks length as u cant pack infinite atoms together... (about 1 * 10^-35 meters)
...wich produces an n-gon with 10^35 surfaces .. so the maximum namber of physically possibly angles would be 180( (360+1)/10^-35 ) - 2) =180(361 * 10^35) - 2) degrees, but it higly depend on whats atoms the circle r made of .. differen atom can be packed differently and depends of energy(temp) and external Pressure) .. or about 6.497999999999999999999999999999999999998 × 10^39 degrees is physically possible, 180(361 * (1/Plancs length) ) - 2) is a more pricise anwer
corrction: the angles (using planks length) adds to: 6.49799999999999999999999999999999999964 × 10^39 degrees...
(in addition: the smalles orbial width of an electron been measered to about 1*10^-22 meters, but nera absolute zero and with high enugh pressure it aproaches 1*10^-35 matematically)
I've had tests to work at nuclear power plants which have questions just as poorly worded.
In one case, it was a 10-question test which you had to pass with at least 80% (8 questions correct). One of the questions was extremely poorly worded so I asked what they meant. However, the proctors said they couldn't talk to me during it. I had just to answer by guessing what they meant to say. After I finished, I went to the overall instructor to explain why I was so frustrated. I indicated what question I hated, and he agreed that the question stunk.
If someone missed it, they would have only had 1 more question they could miss without failing.
This question lacks mathematical rigour and is therefore unanswerable.
The question could have been stated more precisely as "What is the sum of the interior angles in the following shapes?" . If "degrees inside a shape" is taken to mean the sum of interior angles inside the shape, that meaning should hold good for all shapes including the circle for which the correct answer then would be infinity. The example with "present" used three times is a false analogy because the word "present" is used thrice in different contexts. But in the math question, the "degrees inside a shape" although constituting loose language is used just once in the same sense for all shapes. So if the answer for circle is expected to be 360 degrees, it should be 360 degrees for all shapes because then it can be concluded that "degrees inside a shape" is not used in the sense of "sum of interior angles", or else the answers should be 180 degrees, 360 degrees, 540 degrees, and infinity degrees respectively for the triangle, rhombus, pentagon, and the circle using the 180(n-2) formula.
I could be wrong here, but I've always felt that a curve and an infinite number of points approximating a curve are not the same thing. A circle has no interior angles, so why are we trying to add them up? Now the question didn't ask to add up all the interior angles, but it also didn't bother to use proper grammar. "How many degrees inside the following shapes?" Presumable what they meant to write was, 'How many degrees ARE inside the following shapes?'. Since we don't have an objective question to answer, what are all the possible questions that could have been meant by this question (I've tried to list a few):
a) Using the formula 180 degrees (n - 2) = a, where n is the number of sides of a given shape. What is the value of a for each of these shapes?
In this case the answer depends on how many sides you consider a circle to have. If you're taking a purely mathematical approach, a circle has 0 sides, and therefore the answer should be -360 degrees. Otherwise you could take a more philosophical approach in which you could argue for any number of sides. The most obvious arguments being that a circle has either one continuous side or infinitely many sides reaching answers of 180 degrees and infinite degrees respectively.
b) What is the sum of all interior angles of each shape?
In this case the answer depends on how many angles a circle has. As before you can argue for both infinite angles and zero angles, and in the case of infinite angles the answer is the same. However, in the case of zero angles we now reach an answer of 0 degrees.
c) What is the central angle of the following shapes?
Now the answer for each shape is 360 degrees.
d) How many degrees are inside the following shapes?
And now, the answer starts to get real tricky. Standard test-taking practices suggest that if we are going to interpret this question, we need to interpret this question the same way for each shape, although there is no law of the universe that requires us to do this. We could, for example, interpret point a) for the triangle, rhombus, and pentagon, and choose to interpret point c) for the circle. This would give us the answer the school was expecting, but why would you ever assume this is the interpretation the school wanted. Well, when you've gone to school for as long as I have, you just get a sixth sense for this sort of thing.
Or there's my favourite interpretation of this question which is that there are an infinite number of points within each of these shapes and each point has 360 degrees because of course it does, and therefore each shape has infinitely many degrees inside of them because you really should be more precise when wording a test question.
Oh and I also forgot to point out that for options a, b, and d; you can also logically conclude that the answer for a circle is undefined, nonsensical, or meaningless. For example, if you were interpreting the question as use the formula for calculating the sum of all interior angles of a polygon, you could then say that a circle is not a polygon and therefore an answer does not exist or make sense, etc.
But basically, if I had to sum up my opinion: the school is wrong, and this shouldn't be surprising.
This is essentially the same argument that I was making, although instead of saying that it is undefined, I think zero is a perfectly good answer.
I suggest the answer is "undefined", as a circle does not have interior angles the way a polygon does. The central angle is 360°, yes, but that's not the same as a polygon. I guess I agree with ChatGPT?
The answer zero can also be defended.
We don't sum all the 180 deg angles along the straight edges of the polygons, we only aim the angles where straight sides meet at some other angle other than 180.
Two arguments follow on from this point
There are no straight sides on a circle, so there are no angles to sum. The sun of zero of anything is zero.
Alternatively, if we are going to allow "infinity" as the number of "sides" of a circle, that means the internal angle "is" 180 and should be ignored, as with a non-angular node anywhere along the sides of a polygon. We should therefore ignore all these "infinite" numbers of angles.
To be more rigourous, the above argument should be rephrased in terms of the limit as n -> infinity. But that's not how the problem was set ...
A third aspect to consider is that a limit _cannot be used_ to determine the sum of interior angles of the circle.
For example, we could define a sequence of polygons such that their shape approaches that of a triangle as the number of vertices approaches infinity, but where the sum of interior angles is and remains, say, 360 degrees, _even in the limit_ .
[ Start with four vertices A = (2,2), B = (0,2), C = (0,1), D = (1,1). At the k'th iteration, add the vertices Pk = (0, (1/2)^k) and Qk = ((1/2)^k, (1/2)^k). Then the polygons
ABCD
ABC(P1, Q1)D
ABC(P1, P2, Q2, Q1)D
ABC(P1, P2, P3, Q3, Q2, Q1)D
...
ABC(P1, P2, ..., Pn, Qn, ..., Q2, Q1)D
etc.
are all quadrilaterals, each with a sum of interior angles that equals 360 degrees, and hence the limit of the sum of interior angles (as n --> infinity) is also 360 degrees. However, as n approaches infinity, the shape of the polygon approaches that of the triangle ABO (where O = (0,0) is the origin). ]
Would that then mean that the sum of interior angles in a triangle is 360 degrees? No, of course not -- the sum of interior angles in a triangle is always 180 degrees.
This shows that the limit of the sum of interior angles generally cannot be used to determine the sum of interior angles of the limit shape.
@@yurenchu Your proposed construction has a problem with treating the vertices inconsistently - it counts them all for "number of vertices" but only counts four of them for determining angles.
The point - that the limit of a property isn't the same as the corresponding property of the limit - is valid, but the example isn't entirely convincing.
It's less directly related, but I like to use the staircase construction of a diagonal line to establish the principle - a staircase connecting (0,0) and (1,1) has length 2 even as you double the number of steps, but converges pointwise on the straight line connecting the two points, which doesn't have length 2.
The issue that I see is that when questions are grouped together like this, that usually means there's a common trend, or they're all very similar.
Randomly throwing a curveball about radians and center points isn't only a (slightly) different level of work, but completely differs from the rest of the question!
I would have answered infinity or zero for the circle, depending if you consider a circle having zero or infinite corners.
"Studies show that girls and boys with exceptional grades are most likely the teachers favorites." I don't know the exact original sentence, it was in German anyway, but there is a part where a Mathmatician should want to add brackets. It could mean "(girls and boys) with exceptional grades" or "girls and (boys with exceptional grades)". Math is inherently based on logic and should be based on logic always, whereas language is inherently based on chaos and nearly void of logic. That's why the word "infinite" is pronounced different from "in finite". Language does not follow logic, so Math should not follow language like that.
That being said, I had a similar experience in Math once. I was asked to prove n*(n/(n^2-1))^(1/2) = (n+n/(n^2-1))^(1/2), given multiple examples. Thing is, in the examples there is no plus sign on the right side, which I completely forgot about. Basically, if you have a natural number ahead of a fraction, in concrete digits, it means they are added, but with variables, it means they are multiplied.
There is a single, generalized definition that works for all cases and yields the expected results: Follow the outline tangentially. Whenever you reach a corner (i.e., a single point where the path is not differentiable because the left limit is not equal to the right limit), make an instant counterclockwise turn from the previous tangent until you align with the new tangent. This way, you will have turned 180(n-2) degrees traversing a n-gon and 360 degrees traversing a circle, and you can figure out degrees for even more complex shapes.
When I try following those instructions, I get a rotation of 360 degrees for all polygons. For example, with an equilateral triangle, I turn 120 degrees at each corner in order to get from one side to the next.
Interesting that a math question can teach a student so much about life and communication with others.
The question is worded to be technically correct only for the circle, for polygons you have to use your intuition to realize the intended goal is an interior angle sum. Somewhat ironically if anything the questions the dude's daughter got right should be thrown out and the one she got wrong kept in, if you take things super literally anyway..
notice this is true no matter how we draw this triangle...
even if we animate it to funky 8-bit music,
it still adds up to 180°