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The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside.

This can be proved easily by considering a cone as a solid of revolution, but I would like to know if it can be proved or at least visual demonstrated without using calculus.

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+1. I always believed that a rigorous proof required calculus, but I'd love to be shown otherwise. –  Larry Wang Jul 24 '10 at 3:41
    
The Egyptians new how to calculated pyramids. Turned out the shape didn't matter just the base area. Democritus put it together en.wikipedia.org/wiki/Democritus#Mathematics –  Jonathan Fischoff Jul 24 '10 at 4:04
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another twist on this: The other well-known calculus way to find the volume of the cone is to integrate over the disks that make up the cross sectional areas, as you go from the base at 0 to the top at h. The same argument (though it uses calculus) can show that, if you take an arbitrary region in the plane, and first form a "cylinder" of height h from it by extruding it a distance h, then form a "cone" from it by extruding and then tapering linearly, the volume of the resulting "cone" is 1/3 the volume of the resulting "cylinder" (loosely, making solids pointy nicely gives 1/3 the volume). –  Jamie Banks Jul 24 '10 at 4:13
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@Katie: Well it only takes calculus to formally prove it. Imagining a cylinder as an "infinite-a-gon pyramid" is really good for intuitively understanding thas –  Casebash Jul 24 '10 at 6:48

9 Answers 9

up vote 35 down vote accepted

alt text
A visual demonstration for the case of a pyramid with a square base. As Grigory states, Cavalieri's principle can be used to get the formula for the volume of a cone. We just need the base of the square pyramid to have side length $ r\sqrt\pi$. Such a pyramid has volume $\frac13 \cdot h \cdot \pi \cdot r^2. $
alt text
Then the area of the base is clearly the same. The cross-sectional area at distance a from the peak is a simple matter of similar triangles: The radius of the cone's cross section will be $a/h \times r$. The side length of the square pyramid's cross section will be $\frac ah \cdot r\sqrt\pi.$
Once again, we see that the areas must be equal. So by Cavalieri's principle, the cone and square pyramid must have the same volume:$ \frac13\cdot h \cdot \pi \cdot r^2$

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Great animation. I wonder if anyone makes any good (toy) physical models of such... –  Jesse Madnick Jan 16 '13 at 6:04
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But why are there two? –  TonyK Jul 28 '13 at 17:41

You can use Pappus's centroid theorem as in my answer here, but it does not provide much insight.

If instead of a cylinder and a cone, you consider a cube and a square-based pyramid where the "top" vertex of the pyramid (the one opposite the square base) is shifted to be directly above one vertex of the base, you can fit three such pyramids together to form the complete cube. (I've seen this as physical toy/puzzle with three pyramidal pieces and a cubic container.) This may give some insight into the 1/3 "pointy thing rule" (for pointy things with similar, linearly-related cross-sections) that Katie Banks discussed in her comment.

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One can cut a cube into 3 pyramids with square bases -- so for such pyramids the volume is indeed 1/3 hS. And then one uses Cavalieri's principle to prove that the volume of any cone is 1/3 hS.

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This is certainly a non-trivial proof if you have to avoid calculus. See the answer over on the Dr. Math forums.

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Dr. Math's answer smells like calculus to me. –  Larry Wang Jul 24 '10 at 17:35

I just did a demonstration with my class that took about 2 minutes. Granted it was just inductive reasoning but it satisfied the students for now. I had 2 pairs of students come up to the front of the class. One pair had a cone and a cylinder. One pair had a pyramid and a prism. Each pair had solids with a congruent base and height. The person with the cone had to see how many times they could fill the cone with water and fit it into the cylinder. Similarly the person with the pyramid had to see how many times they could fill the pyramid with water and fit it into the prism. Other than ensuring that the cone and the pyrmaid were not overfilled (taking into consideration that the water has a curved skin at the top) the experiment was simple and the demonstration made it easier for the students to remember the relationship. Hope this helps.

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It is because a triangle in a box that has the same height and length is 1/2 if the square because it is in the second dimension so if you move in to the third dimension it will change to 1/3 and so forth.

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This is not much of an argument. Could you provide some algebraic detail? –  Simon Hayward Nov 28 '12 at 21:53
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Yes, it's not clear what the "so forth" means -- what is the fourth dimensional counterpart? –  bryn Dec 1 '12 at 9:30

First You need to understand how to derive the formula to find the volume of apyramid. Once you have this down then it's simple to see why it is $\frac13\pi r^2 h$ because a cone is no more then a circular pyramid.

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imagine a pyramid inside a cube 1 of the point of the pyramid is touching the top face of the cube That one point i just mentioned can go anywhere as long as its on the top face of the cube and still not change the volume. Imagine that the point i just mentioned went to the corner of the cube Cut the top half of that pyramid The top half u just cut would look exactly like the pyramid,except that the volume is exactly 1/8 of the original Now look at the lower half You will notice that u can cut a part of it to get the exact same shape as the top half. Cut it so you have 2 of those small pyramids. The remaining object will have a volume 1/4 of the cube the two small pyramids is 1/8 of the original.since u have 2 of them the two parts combined will be 1/4 of the original pyramid Which means the remaining bit is 3/4 of the orignal pyramid,which is 1/4 of the cube the 2 parts are a 1/3 of the remianing part so if we add them together (1+1/3)/4=1/3 sorry this is confusing but it works :P

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You'd have to use calculus to derive this formula in the correct manner. I have created a playlist that you can watch if you'd like to know exactly how this formula can be derived from first principles. The playlist is almost an hour long:

Finding The Formula For Volumes Of Cones

In this playlist you will learn what slopes are, how to differentiate, how to find areas underneath curves, how to find volumes using co-ordinate axes and then how to find the volumes of cones formula using calculus.

If you have any extra questions regarding these videos, I will reply to them below.

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