The Volume of a Sphere - Numberphile
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- Опубликовано: 19 июн 2021
- Johnny Ball discusses Archimedes and the volume of a sphere.
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"I bet it does" has never worked for me on a math exam 🙁
you have to supply the wooden models and a tub of water
@@letMeSayThatInIrish But wood floats. You need to make the models out of something heavier than a duck at least.
@@fltchr4449 Not all wood floats, quite a lot of the time something made of wood will sink
@@fltchr4449 Very small rocks!
@@hughcaldwell1034 Yes! Those float too.
This guy is the grandpa that tells everybody cool facts and gives his grandchildren candy before going home at a family gathering
"This guy" is a Great British Legend.
@@dorksouls978 i’m happy for that. Also, don’t dishonour him, he’s not a legend, he is THE legend
@@dorksouls978 Leonard Euler was also a great legend. One of the things I consider legendary about him was that was playing with his grandkids when he got a stroke and died.
I would be happy if I could die under similar circumstances! Of course, getting run through by a sword because a soldier was interfering with my mathematics is a close second (which is how Archimedes died, incidentally).
@@sophiacristina lucky
@@sophiacristina aww, thanks! I thankfully have a lovely family, but this comment made my day
Is there any problem Archimedes couldn't solve by throwing it into water?
Sodium.
Well he didn’t have air conditioning for one
I guess even defence of Syracyse went by throwing crew of Roman ship into sea.
If only I could solve my assignments by throwing them into water...
@@krissp8712 well that is doable - all you have to do is write them on something soluble in water and that should do it
That satisfying click sound as the arrive at the end of the proof, coupled with that lovely smile is how any video should end. Simply sublime.
Sounds like the 'Noooice' click
hardly a proof
@@leonhardeuler675 What makes you say that?
@@Bronzescorpion It's based on observations. It's not rigorous.
@@Bronzescorpion There's mutliple steps that need to be made more rogourous. It's not obvious that the cross-section of a sphere and of a cone are the same as a cylinder. He just says "and it turns out it is...". The similarity between a cone and a pyramid is not properly explained. It's clear to anyone who has done calculus but if you have calculus then there are better derivations of the volume of a sphere anyway. Summing the volume using water does not make it clear that this is true for arbitrary radius. I would in fact say that this falls short of an intuitive argument, nevermind a proof. This belongs in the 1% of numberphile videos that I wouldn't show a high school class unfortunately.
I should say, I'm not one for rigour. I wouldn't demand such things of a youtube video. But this isn't helping anyone. I didn't find it satisfying or infortmative or entertaining.
Episode two of 'archimedes came agonisingly close to discovering calculus'
For the other video I completely get what you're saying, but I don't really see it for this video, could you elaborate?
I dont think so. His method here is too crude.
@@erlandochoa8278 I would guess that he’s referencing the fact that Archimedes is summing up infinitely thin cross sectional areas to form a volume, which is conceptually what an integral would do in this case.
@@thethirdjegs What they showed is not what Archimedes would propose as a proof.
He might get the inspiration for the formula like that, but he would then go on and use his rigorous techniques to make an actual proof
He certainly had the concept 2000 years before anyone else. Archimedes used this method of integration for both volume and mass. An astonishing intellect.
This is one of the simplest, yet most satisfying video on the topic, I have ever watched
Which is why it's even more beautiful - the fact that Archimedes was able to prove it creatively, using different techniques.
@Naman The problem with experiment is that it doesn't *prove* the hypothesis. It only shows that the hypothesis is correct *within the precision of the experiment.* That is not at all the same thing.
@Naman Both are equally beautiful.
@Naman It's subjective though. If they think it's beautiful, it's beautiful to them. You can disagree, that doesn't mean that they're wrong.
@Naman don't be so negative, it's Archimedes who did it so don't expect much since our methods aren't available to him, which is the point of Johnny's series regarding how ancient mathematicians discovered facts of geometry.
I recognised that voice immediately. Johnny is a legend.
Johnny ball
Mr. Sins?
Me too. When I was little I used to watch him and that was 35 years ago. He doesn't seem to have aged much in the intervening years. I reckon its because he always talks so gently. A lovely man.
Ditto! What a lovely surprise
"Johnny Ball discusses Archimedes and...."- I like the fanboying of this old man.
Great to see Johnny Ball again. Must be over 30 years since I last saw him on telly. Definitely a fanboy.
He's not an old man. Your age divided by his at any point in your life progressively approaches 1 when plotted. We just have short life spans.
@@xenuno well, one thing I know is that the limit of that function should not exist at infinity cause here both lives are bounded but with different bounds of range.
??
Me: pulls out my trusty bath of water to measure the volume of an object to see if it is equal to the volume of another object
The exam supervisor: **visible confusion**
The little animation of him hopping around cracked me up 😆
4:09
With the little sounds. Amazing.
They should have shown him naked, shouting, "Eureka!" Although that was a different discovery...
The beeps sounded like Popcorn to me (the tune not the food)
Another eureka! moment.
Johnny Ball is a legend...he was a pleasure to watch as a child, and still is. Thanks for getting him into a video, guys!
Johnny Ball taught me a lot via Children's TV. Great to see him back again!
@JohnnyBall Thank you once again. And thank you for getting me to "think of a number" all those years ago. I'm one of the millions of lives you've enriched by making mathematics fun for us at an early age.
@Numberphile Thank you for continuing to platform the all best guest presenters.
extremly interessting. in my school in Germany, we prove the formula of a sphere with the set of Cavalieri but he lived in the 1700 century and that always made me wonder how ancient mathematicians figured it
It's certain Archimedes was way ahead of his peers. We don't know how much ahead. Many of his papers and scientific workings gone to smoke when Alexandria's library disappeared in fire. He knew about Cavalieri's principle and fundamentals of calculus. Some people claim he discovered Newton's laws of mechanics, but that's doubtful
Not the 1700, He lived 287BC , 2,300 years ago. So he was 2000 years ahead of his time.
I've been listening to Johnny Ball tell me things since 1967, and I will never grow tired of it.
Even though he did say cross section, the 2D representation at 1:10 confused me so much, and I was thinking he just meant the width. I get it now, but the 2D representation really threw me off :D
Yes the terms "line" and "cross section" rather than "plane" and "cross-sectional area" didn't help either.
there is still an error in the video at that point... because i can image countless objects that share the same crossection at 3 points but wont add up at all other crossections... so the conclusion is faulty BUT the Water did it right .. its about volume and not 3 crosssections.. its wrong reasoning. i wonder why noone else notices the wrong reasoning?
@@liquidgargoyle8316 It certainly does not make sense to think that just because cross section areas add up at 3 specific levels, the areas will add up at all levels. It does almost seem like Ball suggests Archimedes saw 3 levels and conjectured the sum would be constant for these shapes. In reality you can prove it geometrically. This video is misleading in a lot of ways, as it suggests the water dunking was used as proof when in fact Archimedes had a rigorous geometric reasoning. This is reckless popularization of what is actually a fascinating topic.
@@willjohnston2959 yes ! thanks finally someone agrees :) he questioned if all crossections add up than the frase "i bet it doese" and he dunked into water ... it raises one's hackles when you hear this.. its so faulty reasoning...lol
@@liquidgargoyle8316 I did notice, but you know, actual technical comments in youtube are drowned in the sea of "Wow, that was amazing, what an inspiration!" comments.
I did not know that a cone was 1/3 the volume of the cylinder it occupies. I learned a couple of new things today and I just woke up ☀️
Great way to start the day 😃
Me too
The same applies to any pyramid and its corresponding prism, with one having a third the volume of the other.
Yep, it has a baking application too. Say you have a recipe that calls for 4 teaspoons. Use a tablespoon and "overfill it" so it has a mound on top of it shaped like a cone. Ensure the cone is as tall as the measuring spoon's "bowl", and boom. That is equal to 4 teaspoons.
Comes up a bit when I bake but not every recipe. It does save a bit of time, as it's really easy to replicate
edit: I should mention the bowls of my measuring spoons are cylindrical. Results may vary if you have a more common round one
@@IFearlessINinja There are several variations on the teaspoon (or tablespoon) measure. There's a "level teaspoon" which is the standard measure - fill the bowl of the spoon, but no more, so you have a level surface. There's a "scant teaspoon" which is a bit less than a level teaspoon. There's a "rounded teaspoon" where the substance forms a mound above the level. And a "heaped teaspoon" (or "heaping teaspoon" in the US) where you have as much substance as the bowl of the spoon will carry (if you knock it gently, it'll generally collapse to a rounded teaspoon).
rmsgrey Yes, but the described method is the most simple direct application of the video's explanation
I'm not from the UK, so I've never heard of this man before. Now I'm just obsessed with going after all his work.
I love math. I have a learning disability and can't do it, but I love the elegance & beauty of it.
You will definitely be great at at it and make yourself proud one day!
🔥💯🔥So much this!!!!! 🔥💯🔥
@@meetamisra5505 at the very least he'll be prouder if he tried than just sitting around going "I can't do it"
@@omikronweapon Rude of you to assume he hasn't tried. Dude said he had a learning disability, and most likely he was diagnosed with it after struggling with math throughout grade school. I mean, how else would anyone know he had one? While there may be ways to work around such a disability, in some cases it may be far more effort than it's worth, not to mention you can live a perfectly happy and fulfilling life without being able to do complex math. I wouldn't want to discourage him from trying to overcome his disability (and I sincerely hope he can), but implying that he just hasn't tried is incredibly insensitive.
If you can appreciate the elegance of it, I'm sure you can do it
I wonder what reviewers would say these days if your proof section was "Well, we dumped the thing in water and the level looked right about same-ish. Qed."
LOL, I get what you're saying, but the video is simplified.
The ancient greek mathematicians *loved* their ultra-rigorous proofs - they pretty much invented the concept.
@@jasondoe2596 I broadly agree with you, but I wouldn't call ancient Greek proofs "ultra-rigorous", at least not by modern standards. The proofs in Euclid's Elements have quite a few holes. Though they certainly made amazing progress in rigor.
It's not math, but it is science.
That's not Archimede's proof! That was just a test he made to see if it was worth exploring more his cross section conjecture about the three solids. Then he gave a real mathematical proof on the second half of the video you probably missed :-)
@@gianluca.g I was referring to the "QED" at 1:40.
I've been asking and thinking a lot about how we got the volume of a sphere. I must say this is quite surprising and ingenious. Archimedes really is brilliant.
“How can you prove pi is irrational?”
“Just toss it in that tank of water.”
When I was younger, I thought they discovered Pi by making a cylindrical pool one unit deep and with a radius of one, then poured water into it and measured its volume later lol
@@btat16 I mean, you really could do that, I guess.
Does it get irrationally angry by that, is that how it's proven irrational?
That click at the end, i was just waiting for the "noice" after
i was waiting for "smort"
"Huh, neat."
- Sonic the Hedgehog
If we have a radius of 1 and the centre is at z=0 then the cross section of the cone is πz² while the one of the sphere decreases as π(1-z²), neat :)
Hello I had a question if I am a beginner and didn't study mathematics in high school and now if I want to begin what should I do first?
@@Vizorfam I would recommend that you start with arithmetic. My understanding is that everything else is essentially built upon it.
Think of a Number with the marvellous Mr Ball was my number one, must watch, TV programme as a kid. I attribute my fascination with science and maths to him.
Same here!
So happy to see Johnny again! What a great educator. The "Think" shows were the best children's shows I ever saw 👏
I like how he "clicked" with his tongue at the end.
*CLICK* NOICE!
This is such a terrific explaination!
My former maths teacher always refused to explain to me the formula, because "it's too difficult to explain". Well, it turned out, it isn't.
Your teacher may not have known about this proof. The proper proof involves calculus and is indeed too complex for a school kid.
This isn't a complete proof though. By "crossection", he should have meant area, not length. With area, it isn't obvious it is. You need some simple pythagorean theorem and some simple argebra at minimum to prove its crossection's area.
@@olmostgudinaf8100 You can prove it with just simple pythagorean theorem tho.
2:54 Noice
Be honest, you only got Johnny for this because of his surname, didn't you?
B A L L
lol
YESSIR
Johnny was a regular in British children's television in the 70s and 80. He popularise maths, and made science interesting. I click on the video because Johnny always had interesting things to say.
@@lauraketteridge324 I'm familiar with Johnny's work, I just thought I'd point out the coincidence.
Johnny Ball… my childhood hero! Great man, great to see him on this channel!
The rare time that I actually fully understand one of these videos feels just so wonderful :)
Johnny Ball is the true master. Imagine him and Archimedes having a chat!!
It would have been nice to explain why the sum of the two cross sections is constant. This could have been shown very simply using Pitagora’s theorem to show that, if you put the origin at the centre of the sphere, the triangle with one vertex in the origin, one at (0,h) and the other at z=h on the surface of the sphere, h^2+r^2=1, so the area is πr^2=π(1-z^2).
At the same time the area of the cross section of the cone is πr^2 = πz^2
Therefore their sum is πz^2 + π(1 - z^2) = π(z^2 + 1 - z^2) = π that is constant in z
I think I need a diagram to understand this. I got lost almost immediately. r²+h²=1 sounded like it represents a right triangle, but the hypotenuse of this right triangle doesn't seem to be useful to the goal. It's r units to the right and h units up which would not be on the surface of a sphere. That's how a cylinder would look, though. But then the hypotenuse would also be unhelpful to calculating anything related to the cross-sectional area.
To me, I think you'd want h²+r²=R² where h is distance above origin, r is radius of the circle making up the cross section of the sphere and R is the radius of the sphere. Then the area of the sphere's cross-section would be π(R²-h²)
The cone would have a cross-sectional radius r=h because it linearly increases from 0 at the origin to R at height R so the area of the cone's cross section would be πh² and we still get that the sum would be πR² which is invariant of h.
I suppose that means you used R=1 to save time, but that doesn't track with my picture of the right triangle not lying on the surface of the sphere. I see that yours works, but I can't see why.
Edit: wait, nope, I got it. If R=1 then r is what we want. It's exactly the same. I'm not sure what I was smoking, but it would have been a lot easier if you explained what any of your variables represent.
This explanation takes out so many details that it actually becomes confusing. Why is it that you can extrapolate your knowledge about square piramids to cones? If you think about it it is correct, but this fact alone could be more interesting than most of the rest of the video. Could Archimedes actually prove that the volume of the cylinder equals that of the sphere plus the double cone? He could have, using Pythagoras theorem, it's not that hard. So many questions. Overall this video felt rushed.
Also, what about pi? what about numbers? what about what is mathematics? so many questions unanswered...
This man has one of the best storyteller voices I've heard in a looooong time!
I think Archimedis' thoughts were a bit more sophisticated than "let's throw it into water".
As far as i know he showed that by the Pythagorean theorem the cross sections of the two figures are the same at each height and thus (by assuming Cavalieri's principle) they must occupy equal volume.
It's true that the video doesn't do justice to the topic, but also the way he did it in his mechanical theorems is slightly different than Pythagorean theorem + Cavalieri's principle. Notably, he weighs the slices on a lever. He assumes each slice has mass proportional to it's area and he balances the torques of the slices on the lever.
It's a bit confusing for us, but worth looking into
Exactly. This is how I know this old story.
He was also known to use levers (balance) to prove things.
how is the cross-sectional area sum hold true? cone line is linear but the circle has a curvature
Yes, but because of the fact that the radii have to be squared to get the cross section’s area, the shape of the side view is distorted from linear
Leaving a comment to come back to when somebody answers this.
At height h from the center the cross section of the double cone is π h^2 while the cross section of the sphere is π(r^2 - h^2) by Pythagoras's theorem applied to the triangle [r, h, section].
doesn't hold for the radii but the sum of the areas of the cross sections is indeed constant.
At height h, the sphere's cross section has area pi*(R^2-h^2) by the pythagorean theorem and at height h the cylinder has cross section pi*h^2 because it has inclination 1. The sum is therefore always pi*R^2.
Because the cross-sectional area rises by the square of the radius. If I have 2 same sized discs and I increase the radius of one by 1 unit I would have to decrease the other by more than 1 unit to keep the area sum constant.
Seriously. Bring back Johnny. Don’t get me wrong, numberphiles with JB are amazing, but we need MORE.
This man has an amazing way of speaking. I could listen to him for days.
I saw a cool video about the surface area of a sphere too on 3b1b
the hypothesis at 1:09 can never be correct, because the cross section of the sphere increases from the top in decreasingly smaller steps to 1, while the cross section of
the double pyramid descends linearly from 1 to 0. At points in between the top and the middle the total will add up to more than 1 and less than 1. Or am I missing something?
At height h from the center the cross section of the double cone is π h^2 while the cross section of the sphere is π(r^2 - h^2) by Pythagoras's theorem applied to the triangle [r, h, section].
The double pyramid doesn't descend linearly, since a cross section of the pyramid is a circle. So the area of a cross section is proportional to r^2, not r
It's also non linear for the cone, the cross sectional area is proportional to the *square* of the height of the cone (as measured from the tip of the cone)
I think there was some confusion. The cross section is an area which does indeed satisfy the relation given in the video. But it looks in the animation like they’re comparing widths which as you said don’t add to one
@@Dymodeus1 This is really helpful in understanding what's happening. You need to mentally view this from above, then you can see circles for cross sections, one increasing as the other decreases. Of course the cylinder is constant.
I'm am old enough to remember "Think of a number" on TV - he's brilliant.
1. Johnny Ball is the right guy named to explain this ball problem.
2. He explains the solutions like he's revealing a secret.
2:54 Nice
This is lovely. Im planning to be a math teacher later, im gonna remember this and show it to my students ^^
Don't. This is rubbish. There are far better proofs.
I'm 60 years old. Loved this Guy when I was a kid. Just seen this video and remembered why.
Johnny Ball! If you were a kid in the 80s you'll be freaking out right now. This guy was every kids favourite maths teacher. Absolute ledge! Fun fact: There's a tiny lane in Bristol city centre called Johnny Ball Lane in honour of the great man.
If you don't mind me asking who's on your pfp
Archimedes be like:
Cross section of cylinder is equal to sum of cross sections of sphere and double cones. *Let that sink in*
Can math be also ASMR?
This guy: "I bet it does!"
I stopped my bike ride this morning to watch this video, and it was totally worth it!
Another video that has provided me with that rare feeling of genuine enlightenment, a simple of concise explanation of something that is so clear that you feel like a weight has been lifted from your shoulders, suddenly you can see clearly.
I have a degree in math and I've wondered about this for a very long time
It can also be calculated by integrating the area of the cross sectional circle from -r to r.
@@thistamndypo Yes, but calculus wasn't invented yet when archimedes lived.
can anyone explain to me why the slices between the top and middle would add up to the same value always when the cone has flat sides and the sphere is rounded?
It's areas of circular cross sections that add up. Not lengths. The video was showing a side view, but think of the top view.
Great to see Johnny Ball on Numberphile.
I like this style of video. It's got a simple satisfying point, and gets there without a long ambling story, but does it without rushing.
I don't want this to be the only way you do videos, as sometimes the 20 minute sit and think ambling story is both interesting, and neccessary for the big picture you want to show. But also because I like that I can occasionally make one of the interesting leaps in logic before they tell you.
I like the tongue click near the end
Nice
Nice
Noice
Noice
Noice
Just because the volumes are equal, that doesn't necessarily mean at each level the sums of the cross sections are the same. There are a couple steps missing in that proof.
The cross-sections are irrelevant - as in, you can take them out of the proof and the proof is still valid. It's a leap of logic (that feels counterintuitive and should be wrong) that formulates a hypothesis (the hypothesis being that the volumes are equal), but once that hypothesis is validated, the rest of the proof stands on it, not on the leap of logic. The conclusion has nothing to do with the cross-sections after all, just the volumes.
The volumes being equal, means that (sphere+cone+cone = cyllinder), you can calculate everything except the sphere, so from there you isolate it and become able to calculate it.
(s + 1/3*pi*r*r + 1/3*pi*r*r = 2*pi*r*r)
(s = 2 - 2/3(pi*r*r))
(s = 4/3(pi*r*r))
Yeah that part confuses me, because the sums of the cross sections are not the same, since the variation in cross section of the cone is constant, whereas the cross section of the sphere varies at a much faster rate near the top and bottom.
No, it does mean the sum of cross-section is the same. Volume is height * cross section area, since they have the same height the sum of the cross sections must be equal
@@lumer2b No. If you have the cones flipped so the bases are in the middle and points on top and bottom the volume remains constant but the cross-section clearly cannot. It is indeed an incomplete and misleading proof.
@@taliyahofthenasaaj7570 If it were irrelevant, it should not have been discussed. Yet it was, and then not adequately addressed.
litterally changed my life
"Oh, a 4-minute new numberphile video, I guess I'll have a quick watch"
*After watching*
"Oh the actual video was only 3 minutes..."
Thank you for this! I've long known that the volume of a cone is 1/3 that of a cylinder of the same base and the same height. From the formulae of the volumes of a cylinder (πr^2 h) and sphere (4πr^3/3) it's easy to see that the latter is 2/3 of the former when h=2r. But I've never seen those facts brought together like that.
And an application of Pythagoras's theorem on the radii in each cross-section proves that cylinder = sphere + cone in cross-section area. Neat. But the video should've used that to *prove* the sphere volume formula, rather than just *assuming* that cylinder = sphere + cone in volume.
Here's the missing part of the proof: Let the z-axis be parallel to the cylinder's axis, with z=0 at the sphere's centre. Then the cross-sections at z are circles of radii r (cylinder), sqrt(r^2-z^2) (sphere) and |z| (cone). The sphere cross-section radius formula comes from Pythagoras. So the cross-section areas are πr^2, π(r^2-z^2) and πz^2, QED.
I don’t think there was an assumption. I think they used the same historical argument of Archimedes with the water displacement method. Of course with modern methods we can have more rigorous arguments. The greater assumption in the video I think was making an analogy of a cone/cylinder relationship with a pyramid and prism.
@@stephenbeck7222 The cone = 1/3 cylinder relationship was known to Eudoxus and Euclid earlier, so Archimedes was free to employ it as a given. Certainly this video jumps all over the place and skipped this.
Ancient mathematicians: Uses complex methods to find the volume of a sphere.
Newton: Haha calculus go brrrr.
Is calculus itself not a complex method?
@@Dalenthas
modern day dudes: haha matlab go brrrrr
I'm legit sitting in my room clapping coz this video deserves a round of applause! FINALLY an easy way to understand the formula! Thank you!!
in high school science, I realized I didn't have to memorize everything, only the fundamental concepts, and that I could use them to recreate other concepts as I needed them. Little did I know I had independently stumbled across the concept that I would many years later learn people refer to as "First Principles". So I set about practicing recreating formulas and such from scratch. I actually came up with the equation for the volume of a sphere in this manner, without using calculus. Unfortunately I don't remember the method and logic I used to solve it. Wish I did. Of course at the time I didn't realize it would be something that frustrated me for years to come. If I'd of known then, I would have written down how I solved it. I used only logic and math, no actual physical experiments, and it took me maybe 15min to reason my way through it. I just can't remember where I started and how I reasoned it, but I ended up with the correct answer.
Archimedes Solution is spectacular.
Hey guys, great videos. I happened to be checking my subscriptions feed as this was uploaded, so I thought I would get in early.
One thumbs up due solely from the satisfactory popping noise he made with his mouth at the end. Well done sir.
Well everyday is a school day....used to love Johnny;s TV programs growing up.- still seems sharp as a tack.
I imagine Mrs. Archimedes opening the door to his study and saying 'are you going to mow the lawn today or what?'
'Sorry dear, I need to work out the volume of a sphere'.
The genius part is coming up with the idea of using the cones and cylinder to derive this. How did he think of that?!
well he was archimedes
He pulled his bong apart.
Imagine Euler calling you a genius... damn
For anyone interested, that idea of two shapes whose cross-sections are the same at every height having the same volume is called the Cavalieri's Principle and it is one of those pre-calculus notion that is so close to it
The simplicity and elegance of this proof is mind blowing.
Even he proved mathematics equation by Archimedes principle.bravo!!
Or, as he called it, "by my principle".
Idk it doesn't feel quite rigorous to just measure the volumes, how do we know they're not ever so slightly off?
Here's a different approach:
Write out the cross sectional area of (the bottom half of) both shapes as functions of height. For the sphere, the area is simply pi*R^2 where R can be substituted with sqrt(Rmax^2 - (Rmax-h)^2). The area of the cone's cross sections are even simpler: pi*R^2 where R=Rmax-h. If you add those together, you'll find everything cancels out nicely leaving you with a constant area of pi*Rmax^2
This needs to be practiced in math class. I would have had a better job memorizing the area of a sphere if I got to do hands-on dunking shapes in water
If you put in formula you just get that: given a sphere and a cone of radius r, the section at height x on that sphere will have radius y=sqrt(r^2-x^2). A section at height x on the cone will have radius x. So, we have that pi r^2 = pi sqrt(r^2-x^2)^2+pi x^2=r; r^2 =r^2 - x^2 + x^2;
Great video but I would love an explanation other than "he dumped in the water an they measured the same"
Build a time machine and take it up with Archimedes...
He left out the background which you were supposed to be infused with in school. The textbook story is Archimedes had an epiphany moment one day as he settled into his bath tub, and noticed that the water rose up in the tub as he lowered himself down into it. The story was the tub overflowed and thats when the ahah moment struck him. He was wanting to find the math to the volume of a sphere and couldn't work it out until he noticed that displacement volume was the same no matter the shape of the object submerged. But royalty also wanted a "test" for actual gold content and displacement volume proved to be the key to test for pure gold against facsimile items as its displacement always equals a math of its weight. It weighs 19.3 times the amount of water it will displace. His actual claim to fame might be this gold test or it already existed and he borrowed heavily from it to find the maths for the sphere. At any rate, which ever is the truth, he was a very clever, practical man. As soon as he mentioned Archimedes, I figured water displacement and model dunking would be involved somehow. Exactly how and the relation to cones was what I didn't know.
Well if measuring a volume with water is part of the proof, then you can just directily measure the volume of the sphere with water, right?
2:28 explains it.
They sphere and double-napped cone have to cross-sectionally 'complete' one another. When the cross-section gets to the middle height, the cone is one third of the volume of a little cylinder which encloses it. Therefore, the hemisphere has to complete that little cylinder's volume by being the remaining two thirds of it. Double the hemisphere and we get the formula for the whole sphere.
This video does not prove that the cross-sections do complete one another. It implies that Archimedes just 'bet that it does' and rolled with it.
@@curtiswfranks The narration does imply that sort of loose "lets roll with it" attitude, but Archimedes did actually provide a proof that slices of sphere and cone match up to slices of cylinder. He was not so cavalier. ;-)
Johny Ball...Absolutely loved this man when I was at school. Best ever kids TV show.
Another gem from Archimedes.
That's exactly how my schoolbook told it.
Maybe the volume of a sphere were the friends we made along the way.
Why is everybody writing that now? I understand the joke, but its rather old so why are people writing that rn?
@@doim1676 I've never heard it.
@@0ia
A big rock
your house
🐈
@@alansmithee419 Well, I've heard the ending, "were the friends we made along the way."
Is "Maybe the volume of a sphere were the friends we made along the way" such a popular joke?
I just love the white noise in every video
Please don't ever remove it
I don't know if trig identities existed during Archimedes's time, but if they did then you can use that to prove that the sphere and double-cone will have the same combined cross-sectional area as the cylinder for any point. Instead of labeling points based on their vertical height, we can imagine a unit circle superimposed on each object, and label points based on the angle that the point on their height/cross sectional plane would make with the horizontal and call this x. All of their cross sectional areas are circles, so we just need to find the specific radius for each value x. For the cylinder this doesn't really matter since the radius is always the same (we can call this R) so the area is always pi*R^2. For the circle, the radius is how far the edge is from the vertical axis, which is the same as cosx times the overall radius, or Rcosx, so the area is pi*R^2cos^2x. The sides of the double-cone slope up along the standard y=x line, so the radius will always be the same as the height. Again putting this in terms of the angle x, the height is sinx times the overall radius, or Rsinx, so the area is pi*R^2sin^2x. If the area of the cylinder is equal to the area of the other two combined for any value x, then you can write that as pi*R^2 = pi*R^2cos^2x + pi*R^2sin^2x. Since every term has piR^2, you can cancel those out to get 1=cos^x+sin^2x, which is a famous trig identity (you can prove this by picking a random point on a unit circle, drawing a right triangle that connects that point to the x axis and the origin, and then doing the Pythagorean theorem on that triangle). Since this proves that for every cross section, their two areas sum to that of the cylinder, you can also say the same about their total volumes.
I know this kind of goes against the whole idea of doing math very simply without any of our modern tools, but I still found it cool as a fairly easy way to show that their volumes are equal without having to dunk the objects in water ourselves
If I wanted to know about a sphere, I'd have asked a ball.
I wouldve never expected a mathematician to be like "wait lets just do it irl" and that was rigorous enough to prove it haha
Lots of math is ‘let’s just do it’ by running every possible case through a computer, proof by exhaustion or proof by just trying it until it works, no matter how many super computers it takes. See the Numberphile videos over the years on 17, 33, 42, and 3 as the sum of cubes for an example of significant math (I.e. professional organizations investing a bunch of money) motivated by some RUclips videos.
Archimedes was quite the polymath. A profoundly intelligent ancient person.
He was more of a fluid dynamicist than a mathematician. He just needed math for his physics that didn't exist yet, slash was a curious guy
I love this guy. His voice is so soothing
Johnny Ball? I’m instantly transported back to my childhood. 😀
I bet he didn't throw it into the water. I bet he knew about Cavalieri's principle
Sphere!!!!
A crumb of serotonin: Archimedes dancing @1:35
1:40
Getting to the q.e.d. point is just pure happiness
🌊 water done it again ⛲ let's drink to that 🥛
dont tell a flat earther :P
But it's not a real proof, is it. It would not hold up in the real math.
More Johnny please!
Yes, a new Numberphile video ftg. Johnny Ball!!
Funny words mr magic man
I feel like this was cut out of a larger video, and this video suffered for it. I know how to derive the volume of a sphere and even I had no idea what was going on :/
I saw other videos of derivation of volumr of a sphere but this is the best .
Wow thank you for this. It's my first time knowing this solution. All I know is how to get the volume of sphere thru calculus.