How to Calculate Pi, Archimedes' Method
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- Опубликовано: 13 мар 2013
- mathematicsonline.etsy.com
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Using Archimedes' method of exhaustion we can derive a formula that approximates the value of π.
Him realising that the more sides your polygon has the better the approximation is basically the true discovery of integral calculus.
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Now the problem is: how do you compute sin(180°/n)?
Presumably, you construct an enormous right-angled-triangle with angle theta and take the ratio of the opposite to the hypotenuse.
That's exactly my question... sin is defined in terms pi :/
you would have to write taylor series for it (no pi needed). but to use it you would have to convert 180 degrees to radians (pi/2). so the video is pointless
tampicokid i meant you could write the thaylor series for sin(x). By "it" I meant any x. So you can take sines of any angle without knowing pi.
The problem here is that we can't find the angle they want us to input.
tampicokid You can calculate sine of any number i give you. sin(1) sin(2) sin(34). where 1 is 1 radian, 2 is 2 radians.
A circle is usually defined as having 0 sides when, in fact, it has infinite.
Austin Edward those two ideas are practically the same. a circle literally is the collection of points a given distance from another point, on a plane.
@@rH441000 a "side" is a "line segment that joins two vertices in a shape or two-dimensional figure." If a circle is a two dimensional figure that is defined by its vertices, then it MUST have sides.
a circle only practically has infinite sides. mathematically it has zero sides because no matter how closely you look a piece of circle, that piece will always seem an arch and never a "line"
It is just a matter of classification, I don't think it has much difference if you either define it as 0 or infinite. It is a very unique shape after all. Unless you are doing math for the sake of math, called pure mathematics-of which actually ended up having very useful physical implications such as water behavior- utility is much more important than just an issue of classification of a rare case, at least that is what I think
@@johnjordan3552 it has zero sides. period.
Thank you for this wonderfully explanatory video. I am not a mathematician, but over the years I have come to love math. Lately I've been wondering how do we know that the ratio of the circumference of a circle to its diameter is Pi, an irrational number that goes on into infinity without repeating? After all, we couldn't hope to measure it more than to a handful of numbers after the decimal point. The video does a great job for fools like me even if it's not the exact method Archimedes used. Thank you.
The music is Cello Suite No.1 - Prelude, thanks for watching.
I've been looking for a good derivation for at least 1hour now and this one is perfect.
Clearly explained, set by step, with an accurate and clear diagram. Excellent video. Thanks.
😢
3:48 3:50 3:50 3:51 3:51
mathematiconline, thank you so much for your videos. You're a great teacher, and your videos make learning math enjoyable.
The most agreeable definition of pie I have heard so far
"How did we get pi?
Remember that the radius of a circle is equal to 2(pi)r"
Weird flex but ok
It actually makes sense see his previous video
It's high school maths
Pi = 4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/15...)
Not weird. It's basically a pi to 2r ratio, like, "how many times pi would fit in 2r. Pi is basically X here. I don't know how many decimals they had and ended up being correct. (Quite sure they had a better method than calculation by observation by using a thread"). Ambiguous sentence...
2πr
Your videos are awesome and very informative and are on a different level from most explanations, Thank You.
WOW, Great video with understandable math!! Thanks! I could never have believen that it can be calculated so easily!
i just doon't know what sin and theta is but i can surely tell he explained extremely well
This method, by using inscribed polygons, guarantees that each result will be < π ; that is, it approaches the desired limit from below.
If you instead use circumscribed polygons, each result is > π ; that is, it approaches the desired limit from above.
When you do the trigonometry for that, you find that you simply replace the "sin" with "tan":
π = lim [n→∞] n tan(180º/n)
If you want to get a bit more sophisticated, you can show that the limit of the ratio of errors, tan-to-sin, is 2, so that a weighted average,
(n/3) [ 2 sin(180º/n) + tan(180º/n) ]
will converge to π faster than either sequence alone.
sir, could you please tell me how it,'s possible that
sin theta is equal to the side of the polygon (2a=2a)while both have a different angles
Not exactly sure what you're asking, but start at 1:50 in the video, and onward to 3:20 or so.
The circle has radius r = ½, which is also the hypotenuse, c, of right triangle ADB, with right angle at D.
c = ½
BD = a = (½ the side of a regular polygon of n sides) = ½ s
θ = ∠BAD = ½ the central angle of the polygon; so
θ = ½·2π/n = π/n
Now you know that, in right triangle ADB,
a = c sinθ = ½ sinθ
So:
sinθ = 2a = s = the side of the polygon
So what things are you asking about, that have "different angles"?
+ C Noone
Well, you *can* state that; but you'd be wrong.
In particular, Archimedes, who as you say, didn't carry all the way to the end (that is, to infinity!), his procedure for finding π ; he *did*, however, demonstrate that it lies between these lower and upper bounds:
3¹⁰/₇₁ < π < 3¹/₇ ; that is,
3.140845... < π < 3.142857...
It follows then, that π < 3.1446...
So The Sumerians And Archimedes (Proposition 1) and 3 x r 2: Are wrong Despite all three being in total agreement as to having 10, 800 square centimetres to the circle's area,
I really don't think so and perhaps you would also like to try and disprove my twelve steps to the sphere?
Reference Estimating the wealth; Encyclopedia Britannica.
A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value, which would be correct if π is estimated as 3.
Quote: *which would be correct if π is estimated as 3*
The sheer lack of insight conveyed in this comment is truly astounding.
The Sumerians were masters of mathematics, straight linear geometry, and curved differential geometry millennia before the Greeks came along, and it is they who using their differential geometry skills, were the inventors of clocks and time.
Fundamentally
A circle is a round shape, whose curvature consists of a continuing series of points equidistant from a fixed point (it's centre).
The curved edge length of a round shape is a continuing series of points equidistant from a fixed point (it's centre).
The circumference of a circle is the width (area) of a "radiated" drawn line, that serves to surround and enclose the radiated shape (area) of the circle.
Oxford English Dictionary
Circle: a round plane figure whose boundary (the circumference).....consists of points equidistant from a fixed point (the centre).
The first part of this dictionary definition is based in/on the Euclidean linear thinking of more than two millennia ago.
The second part of this dictionary definition is more on par with modern day physics, whereby it can be understood that the equidistant points to the circles centre are spatial points not linear.
Circles shapes and bodies do not have any boundary lines as per Euclidean thinking, all circles have a quantity of area. and all bodies have a quantity of volume.
However, i.e. when we look at a dinner plate from a Euclidean perspective we see it outlined against its background, just as we perceive a silhouette being outlined against a lighted background.
Whereas, when we look at a dinner plate from the physic's perspective, we are aware that there is no outline as such, but rather that the denser atomic structure and reflecting the colour of the plate, is simply standing out from the less dense surrounding atmosphere.
As opposed to nature, in order to make a circle we have to use a drawing compass to radiate a circle, while simultaneously drawing a line around it to enclose it, and so define the limits of its area.
However Mother nature has no need, she simply radiates unbounded circles and cycles (continuum's) of energy conversions in motion, ad-infinitude.
THE LENGTH OF A CIRCLES EDGE
Using a 120-centimetre length of diameter multiply this by 3
1. The circle's edge length is 360 cm long
2. The circle's edge has 360 degrees of subdivision
3. The circle's edge has 360 degrees and each degree is 1 centimetre long
SUMERIAN METHOD FOR CALCULATING THE AREA OF A CIRCLE
Using a 120-centimetre length of diameter multiply this by 3
1. The Circles Edge is 360 cm long
2. Multiply the 360 centimetres "Edge Length" by itself = 129, 600 square centimetres
3. Divide 129, 600 by 12 =
10, 800 Square Centimetres to the Area of the Circle
ARCHIMEDES: PROPOSITION 1.
The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.
Archimedes Triangle
The Circle in question has a 120-centimetre Diameter length
1. The base right-angle is equal to the radius of 60 centimetres
2. The area of the circle is equal to the above right-angle triangle, which has one side that is equal to the 60-centimetre radius, and the other to the 360-centimetre circumference of the circle
3. The 360-centimetre height of the right-angle is equal to 6 x the 60-centimetre radius length
4. (1r) 60 centimetres x (6r) 360 centimetres is 21, 600 square centimetres the area of the rectangle
5. Half of the rectangle is 10, 800 square centimetres
6. The area of the triangle is half of the 1r x 6r rectangle
7. Half of the 1r x 6r rectangle is 1r x 3r
8. (1r) 60 centimeters x (3r) 180 centimeters =
10, 800 square centimeters
THREE TIMES THE RADIUS SQUARED
1. The Diameter of the Circle is 120 centimetres
2. The diameter x 120 centimetres gives, 14, 400 square centimetres to the square of the diameter
3. The 60-centimetre radius x 60 centimetres gives, 3, 600 square centimetres to the square of the radius
4. The square of the radius x 3 gives, 10, 800 square centimetres to the area of the Circle
SUMERIAN AREA: 10, 800 square centimetres
ARCHIMEDEAN AREA 10, 800 square centimetres
THREE TIMES THE RADIUS SQUARED: 10, 800 square centimetres
Twelve Steps From The Cube, To The Sphere (Volume & Surface Area)
Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube.
1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimetre’.
2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 sq cm.
3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.
4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm.
5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.
6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.
7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm.
8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.
CYLINDER TO SPHERE
9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.
10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.
11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm.
12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere
Confirmation by Weight
Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grammes, prior to it being turned on a wood lathe into the shape of a sphere
The Cylinder of the Cube would weigh 120 grammes
The waste wood shavings would weigh 40 grammes
Given that the Cylinder weighed 120 grammes
The waste wood shavings would weigh 30 grammes.
Note: And ironically you can also obtain this same result by volume, using Archimedes Principle.
www.fromthecircletothesphere.net
Now all you have to do is figure out how to calculate sine.
Easy.
You start with an angle that has a well-known sine and cosine, like
n = 6
180º/n = 30º
sin30º = ½
cos30º = ½√3
and you double n, halving the angle, using the half-angle formulas:
sin(½θ) = √[½(1-cosθ)]
cos(½θ) = √[½(1+cosθ)]
That's one way, at least.
ffggddss Now you need to figure out how to work out cosine
No, that's already there - look again.
You start out with a known sine and cosine, and at each step, you have a formula to find the next sine and cosine from the cosine you already have.
SnoopyDoo Taylor Series, as well as nature of circle. We know the properties of a circle circumference, but we are looking for its value. It isnt circular.
No its not like that it is all we have to do was to follow the damn train CJ
mathemental used pi to determine sine values in order to determine the value of pi,stay tuned next week when he and charles darwin solve the chicken and the egg paradox
used pi to find the sine,how is that? thanks
@@josepeixoto3715 The question here is how, at its simplest would you calc pi with pen and paper. Archy did not use trig function at all, because it is not necessary.
Trig functions inherently have pi contained in them. Lets say, with pen/paper you are going to calc the sine of 15 deg. You could do it with the power series x - x^3 + x^5 -x^7......... x is the 15 deg angle expressed in radian. Radian is essentially composed of pi. For instance to express your 15 deg in radian, you first must multiply 15 deg by pi/180 deg. Notice you must already know pi.
Maybe this was poorly said, but it demos that using trig function to find pi, is simply employing pi to calc pi
@@ronalddump4061 No I get what you're saying. This exactly what I've been looking into lately. Everything I research is using pi to explain how we get pi like that. It doesn't make sense.
best outro for a youtube video I have ever seen in my life
By using Calculus, you can turn the sin function into an infinite series (Which I believe was the original method). Integrals are great a summing an infinite number of 'zeros' to get a constant.
Do we have some special triangles for calculating very small values of sin(theta) ?
Great explanation, very intuitive and fast. Good job
This channel is criminally underrated
Thanks for your insight!
great video .it helped me a lot to clear my doubt.the way of illustrating was awesome.thankyou
Excellent. Wonderfully clear presentation. Thanks.
I'm wondering the exact same thing. And also the function Sin(θ)= θ for small angles doesn't work, since we can't calculate the angle in radials. (you need the value of Pi)
Can you tell me which the sequence that approaches pi from above( the genuinely decline function)
Thank you for elaborate the method of π in such a easy manner
Question, if you use radians and not degrees, then, because we already know a circle with radius=1/2 has circumference=pi; then the (length of a side of the polygon)=sin(theta) and (theta)=(pi/2n) ---- which is wrong. Theta=(pi/n). If you're thinking of it in degrees you must divide by two, but if you're thinking in arc-lengths there's no need, but then what (theta) are we measuring? What am I missing?
Excellent explanation
i liked this only for showing some logic. (though i had to think it through a bit) here's it in simplest explanation.
1. split a circle into a number of triangle segments (360/n) 2. half into right triangle (180/n) (to get the opposite length with sin)
3. you would double output for both halves, but sin uses radius 1 (meaning diameter 2) so you would also halve to correct that.
4. you have the segment perimeter! tot them all up for an approximate perimeter! (this cheat method wasn't how we got pi btw)
Very insightful video. Learned something new. Thanks for sharing
So is this why pie has no end in figures? Because it's an approximate estimate the the closer you refine it?
This may have been how Archimedes calculated Pi, but I don't think Archimedes had algebra to work with.
This is not the corect solution of pi is equal to 3.14 or 22÷7
This is not the corect solution of pi is equal to 3.14 or 22÷7
Shankar Sanap no thats only approximate
@@shankarsanap4724 dun dun dutsch!!
What programs or software you use to make this great video?
What about the upper bound that archimedes used? I remember learning that archimedes drew a figure inside and outside a circle of radius 1/2 in order get a more accurate answer.
By the way, if this is the method he used to calculate pi, then it would be similar to rule within limits of trigonometric functions.
Use the tangents, instead of c=1/2 we have b=1/2. So tan∆=2a. This gives π=n×tan(∆/n).
whoever came up with this method is really clever, should get the fields medal
Many people are saying that this is not correct , I don't know if it's true or not but thanks for making efforts to trying to help us please make a video on the true method this was a nice video.
It, in fact, is NOT correct! Because, simply put, Archy did not use trig function to calc pi
Wouldn’t using tangent instead of sine produce the same result?
i want to ask, HOW did Archimedes know in the first place that a circle's circumference is 2pi*r??? You guys mention as if this equation is something all kids BC knew. Archimedes needs to get PI first before he could write down the equation and you guys mess it all up by mentioning the equation first.
Because Archimedes based his stuff of and Indian mathematician called ArryaBath!
rawan hs 2pi*r = pi*d meaning pi is just the ratio of any circle's circumference divided by its diameter. So just take string, measure its length then make it circular and measure its diameter and divide them.
Because thats how pi is DEFINED. The ratio between the circumpherence(C) and the diameter(d). So:
C/d=pi
C=pi×d
d=2r (by definition), so
C=2×pi×r
It was observed that circumference of circle increased linearly with increase in radius/diameter, so that constant was named pi.
B8. RV
What's the music you used in the ending? I know that's a Bach piece, but I'm not sure what the title is...
Can you find the limit as n approaches infinity?
Can we use calculus instead of putting a large amount of number to calculate pi???
why circumference of the circle not measured with a rope directly?
Excellent video! I enjoyed this, thank you :)
Thank you for this. I have a geometry test tomorrow and our
teacher never teaches us anything
LOL What kind of teacher is that?
Triangle ABC is an isosceles triangle, since AB and AC are the radii of the same circle.
For any isosceles triangle, the median from the symmetry vertex is the bisector of that vertex, as well as the it being perpendicular to the opposing side.
This is easily demonstrated by using reductio ad absurdum and falsifying that 2 sides are of equal length, which was the initial premise.
best explanation so far
1:48 what do you mean incidentally? is there a proof for this?
nice! this is a great idea to find Pi.
how to valculate and doer what value ogf sone 180°
By that logic calculating pi precisely be equivalent to dividing by 0 or a divergent infinite sum. That makes more more intuitive sense of euler’s infinite series calculations ending up with pi
How you got answer whenever you divided a number by 180 ??
its been said already, but did he have the formula for the circumference of a circle before his proof? how could they know that it was 2(pi)r and not know what pi was?
At 1:40 line AD actually bisects angle BAC. Therefore angle BAD (which is angle theta) is half of angle BAC. I'm sure there's some way of proving this, but its tedious.
It's because triangle ABC is isosceles (AB = AC = radius = 1/2). In isosceles triangles, the altitude to the base is also the angle bisector. Therefore AD is the angle bisector of BAC, and thus BAD is half of BAC.
Great explanation!
Did Archimedes have trig tables available? I thought he strictly used Pythagoras / square root calculations
How do you calculate the value of sin(x) without knowing pi to begin with? all of the calculations I know of involve expressing your angle in radians.
Saying pi = the limit as n goes to infinity of n*sin(pi/n), while true, hasn't helped me in anyway calculate the value of pi.
Any or all circles it's half circumference devided by reduce gives (pi) 3,14285714....which associated by 180deg.which put by us (so any angle) could calculated
Could you represent pi as a limit as n approaches infinity in this case?
More surprising is to notice n approaches infinity and sin of an angle approaching zero! Pi equals to infinity times sin of zero!
very informative! thank you.
π is the result of the division between the greek words ΩΚΕΑΝΟΣ (ocean) and and ΝΕΙΛΟΣ ( Nile river) when you transform them into numbers according to the ancient greek numbering system where Ω=800 etc. It is very amazing how you may realize the real world on a different basis. Imagine that this division is older then 5000 years.
Tf you on abt??
I Have A Doubt In This Question Is That You Say Sin Theta is opposite/hypotenuse i.e a/1/2 But Is Sin Theta Is Not equal To Perpendicular/hypotenuse that Will Be Equal To In This Case Will Be B/1/2
PLZ CORRECT ME
then why does the answer of the limit when n tends to infinity is 180 and not pi ? :( I calculated it
Fantastic !! Thanx a lot for the video... Have a great Life !!
Hey but it only trys to get to pi as a polygon can never be a circle in true form ...
Wait..Didn't Archimedes use another approach? BECAUSE during Archimedes' time, the sine function isn't there until about 100 years later. But still, I could be wrong. Please correct me if I am.
Excellent! ... but how to prove it is 22/7 ?
Very Informative video
How did he arrived on C=2(pi)r?
rayfabros he didn't really said, I want to find Pi, he just wanted to discover the relation between the diameter of a circle and it's perimeter such as, (perimeter÷diameter)=x
I actually found this on my own feeling I had an inadiquate understanding of pi so that means I’m as smart as Archimedes the only difference…
this guy did it 2300 years ago with a quill and paper no calculators no graphing calculators and literally no teaching using his own number system which most of modern society based off what a legend
Hi, I was messing around with factorials yesterday. And came up with a formula for pi. I was wondering how many digits this formula is accurate to. (-1/2)!(2)
exactly.
Syndee Gaming How did you come up with this?
I was messing about with factorials. as I said
You're using the gama function to find (-1/2)! ?
How was Archimedes using the sine function if the sine function was developed in the 6th century AD?
bruh i am just year 9 and i work out this just when I learnt trigonometry, but I used x*tan(180/x) instead of x*sin(180/x) is it actually that hard to work out this?
FINALLY,i get it, Archi was a thinker... thanks
love that ending!
song at end?
what program are you using
By far the best demo up to now!
wow really impressive method
0:34 Thank you for adding that.
Nice explanation
I swear to you that i was literally just thinking about that exact method when i saw the demonstration of why the sum of all angles in a convex poligon is equal to 180*(n-2). I went to youtube just to realize that archimedes already discovered about it two thousand years ago, lol
Great to know this!
How we know that what to put the value of n
it's mind boggling!
Have a any formula without sin, cos, tan function
why is theta equal to one half of one of the angles?
Just figured out this formula independently before finding this video.
It's amazingly simple, yet all we ever hear about is that ugly and cumbersome "pi=(4/1)-(4/3)+(4/5)..." or "throw a bunch of stick at horizontal lines" or some crap like that.
Using algebra for method pre algebra?
Ramanujan has calculated the value of pie accurately using a more appropriate method
🙋🙋🙋🙋🙋🙋👌👍👍👍👍👍👍
You proved that:
π= n.sin(π/n), as n goes to infinity
which leads to:
π/n =sin(π/n), as n goes to infinity
which is true for any value of π,
and it does not need to be proven, because the opposite side of the angle coincide with the opposite arc as the angle approaces zero.
I think the ancients calculated the value of π experimentally.
Hm why is this just like the limit as n approaches infinity of n*sin(pi/n)? Oh wait pi=180
thank you, I love you my friend
Try this: The cord length of 1/60° (minute of angle) included angle in inches. At exactly 300 yards (10800) inches. sin 1/120° x 10800 x 2 = PI.....My calculator rounds to 9 digits, excel can round to 100 digits.
So, the exact value for pi would be: *∞*Sin(180/∞)*
Retro Gaming - Clash Of Clans i also got that on my head but on the other hand it wouldn't really make sense because infinity times something is always equal to infinity as for the 180 something devided by infinity will always add up to infinity.
Infinite is not a number, it's a concept. But I guess you could re-write that as a limit of x approaching infinite.
@@CMAR872 and you will get an (infinite/infinite)*sin(180) which is undefined ..
Ali Alhasan that’s why it’s a limit though you’re not actually plugging infinity in because that’s clearly impossible
Can any one help me?, the numbers at the beginning aren't right, by putting them in calculator the results are different!
LOl Why the hell is circumference always defined as to = 2*Radius*(pi) instead of just making it simpler C = Diameter * pi.. its the same thing but removes an extra step
I'm inputing 1 for the radius instead of 1/2 and doing the other calculations as is and I'm not getting the right answer FML
@John Stuart are you suggesting Archimedes was stupid?
Given that Archimedes first proposition in regard to the area of a circle is correct, and serves to contradict the formula Pi, no he certainly was not.
Which serves to beg the question after more than two millennia of human history, and in more recent times the decimal/digital bastardization, was the formula his?
No matter the truth will always out, sooner or later.
Originally Pi read as being
Pi = 22/7 as an improper fraction, or 3 whole units and 1/7th of 1 whole unit
Or
3 whole diameter units with 1/7th of one whole diameter unit remaining
Proof
THE LENGTH OF ONE DEGREE OF A CIRCLE
Reference: “Estimating The Wealth” Encyclopedia Britannica.
A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value.
Given a Square measuring 120 centimeter’s times 120 centimeter’s
1. Use one right angle of the 120 x 120 cm square as a Diameter Line
2. Multiply the 120 cm Diameter Line by 3
3. The Circles Circumference will measure 360 Centimeter’s in length
4. A Circle has 360 Degrees to its circumferential length; therefore each degree is 1 centimeter in length.
5. Therefore all Circles have a circumferential that is exactly three times its Diameter Line length;
The Ancient Sumerian masters of geometry and mathematics defined this empirical reality, more than 2000 years before the plagiarizing Greeks.
However one question for me does remain begging, did they also manage to achieve this?
Twelve Steps From The Cube, To The Sphere
Calculating the surface area and volume of a 6 centimeter diameter sphere, obtained from a 6 centimeter cube.
1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimeter’.
2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm.
3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.
4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity of the cube = 54 cubic cm.
5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.
6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.
7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm.
8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.
CYLINDER TO SPHERE
9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.
10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.
11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm.
12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere
Confirmation by Weight
Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grams, prior to it being turned on a wood lathe into the shape of a sphere
The Cylinder of the Cube would weigh 120 grams
The waste wood shavings would weigh 40 grams
Given that the Cylinder weighed 120 grams
The waste wood shavings would weigh 30 grams.
Note: And ironically you can also obtain this same result by volume, using Archimedes Principle.
As To The Proofs OF The Pythagoras Theorem
In any right triangle the sum of the square on the hypotenuse, is equal to the sum of the squares on the other two sides.
Incorrect
Given a right triangle whereby both the base line, and the vertical line of the triangle each measuring 12 units.
The sum of the two squares will be 288 squares
The sum of the square on the hypotenuse measuring 17 units will be 289 squares.
One square greater than the sum of the squares, on the other two sides.
n=1, 1sin(180°÷1)not equal to pi value
Any have a condition for n?
Yes there is.
Can you draw a closed figure by one line segment?
You require at least 3 line segment.
So n must be equal to or greater than 3.