Explaining why volume of cone is a third of cylinder

Question

I came across this video explaining to kids why the volume of a cone is a third of the cylinder of same cross-sectional radius and height. Essentially the explainer presents pre-created cylindrical and conical objects with equal dimensions, and says it takes three cones to carry the same amount of water as the cylinder.

Typically this video is intended for pre-calculus students, aged around 13-14. I understand that calculus is necessary to rigorously prove the statement. I also understand that sometimes there are different varieties of "proofs" - geometric proofs or proofs by construction. Does this explanation fall into an acceptable category of proof? Or is it simply nothing more than a mnemonic to remember the volume formulas?

Perhaps first consider the two-dimensional analog: the area of a triangle is half of the rectangle with the same base and height. — Gerald Edgar, May 02 '22 at 11:35
The thought that struck me when watching this video, with its repeated instruction to try the experiment at home with different cone/cylinder pairs was, "who has a bunch of cones and cylinders with equal base and height lying around the house." I think the video would be improved by the addition of a companion video explaining how to make the models needed to do the experiments. — Will Orrick, May 02 '22 at 13:46
There is a great description of a lesson on this subject over at math.SE https://math.stackexchange.com/a/29570/448 — David E Speyer, May 02 '22 at 16:40
@WillOrrick: Nowadays, anyone with a 3D printer easily has them. :-) — R.. GitHub STOP HELPING ICE, May 02 '22 at 21:33
@GeraldEdgar growing up this personally was never useful for me. Knowing that 2D gives 1/2, my intuition would've suggested 3D either also uses 1/2, or is (1/2)^2 = 1/4. Or maybe even (1/2)^(3/2). I never would've guessed 1/3. Not that my anecdotal experience holds much weight, but the thought experiment certainly might be counterproductive for some students. — Drake P, May 03 '22 at 01:00
@DrakeP Intuition sometimes has to be guided. I think it's a real pity that the common geometric picture explaining the decomposition of the square into two triangles, the cube into three pyramids, and the $n$-cube into $n$ $n$-pyramids isn't more widely disseminated. This has the further nice feature of making $\int_0^a x^{n-1},dx=\frac{1}{n}a^n$ for natural number $n$ geometrically clear. See Steven Gubkin's answer and my comment to that answer. — Will Orrick, May 03 '22 at 15:15
@Drake Fun fact: the constant is 1/dimension. So if you compare the volume of a 4-dimensional cone to a 4-dimensional cylinder, the constant is 1/4. — quarague, May 03 '22 at 15:33
@R..GitHubSTOPHELPINGICE In addition to that, there are companies that sell such items for educational purposes. Usually they sell mostly to schools, but you can get them at home as well. (Helpful for people without access to a 3D printer - which is still most of us at this point.) — Darrel Hoffman, May 04 '22 at 13:55
@DarrelHoffman: Probably a good choice as long as the prices aren't too high; if not, services to order 3D prints are widespread now and low-material-mass quick-to-print objects like these should at least in principle be cheap to order. — R.. GitHub STOP HELPING ICE, May 04 '22 at 14:09

score 30 · Accepted Answer · answered May 02 '22 at 11:18

30

This is an experiment which can lead you to guess that the volume of a cone is approximately $\frac{1}{3}$ the volume of a cylinder with the same base and height. It is not a proof in any sense of the word.

If mathematics were a science then we might have a "theory of the volume of cones" which would be confirmed by repeated experimentation with more and more exacting experimental set ups. However, mathematics is not a science: it is a deductive art. This is, I think, a valuable conversation to have with students.

It is possible to justify this fact without Calculus if you accept Cavalieri's principle.

First make a special pyramid. Use an $a$ by $a$ square pyramid with height $a$, where the apex is located directly over one corner of the base.

https://www.polyhedra.net/en/model.php?name-en=three-pyramids-that-form-a-cube

You can see that 3 such pyramids form a cube, so the volume of this square based pyramid is $\frac{1}{3}a^3$.

Now scaling by $\frac{h}{a}$ vertically gives that the "stretched" pyramid with square base $a$ and height $h$ has volume $\frac{1}{3}a^2h$.

Now take any arbitrary cone with base of area $B$ and height $h$. Compare this to the pyramid whose base is a $\sqrt{B}$ by $\sqrt{B}$ square and whose height is $h$. Using similarity, you can show that these have equal area cross sections. So by Cavalieri's principle, they have equal volume. Thus the volume of the cone is $\frac{1}{3}Bh$.

answered May 02 '22 at 11:18

Steven Gubkin

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This approach shows the origin of $1/3$ at the very beginning, which I think is great - a student may be used to the $1/2$ in $2D$ are formulas, so that's the primary point to address, in my opinion. Does the 3-pyramids-construction generalize to higher dimensions? – Michał Miśkiewicz May 02 '22 at 13:06
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@MichałMiśkiewicz Of course. I made some pictures that, I think, give the general idea. The link is here. I'm not so good with graphics so you may have to fiddle a bit with the animation controls to get a good view. You can stop/restart, speed up/slow down, rotate, and resize the images. – Will Orrick May 02 '22 at 13:19
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By the way, "of course" doesn't mean I saw this immediately the first time I thought about it. It means that after I understood it, I slapped my forehead and said, "of course!" – Will Orrick May 02 '22 at 13:35
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(+1) Great answer, and especially for the idea of a "theory of the volume of cones." At first, I thought, "Yeah, that's a good conversation to have with students if they're learning volume formulas in this way." But also, this might be my go-to answer for "How is math different from science?" I think going into great detail to describe the "more and more exacting experimental set ups" required to develop the great "theory of the volume of cones" will really illustrate the point :-) – Brendan W. Sullivan May 03 '22 at 02:09
@MichałMiśkiewicz It does generalize in the sense that in 4 dimensions the constant is 1/4, in 5 it is 1/5th and so on. I think you can cut an n-dimensional hypercube into n hyperpyramids suitably analogous to the 3d example but I would have to draw and think about it some more. – quarague May 03 '22 at 15:37
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Thanks to both of you! @WillOrrick's pictures suggested to me that in $n$ dimensions you can choose two opposite vertices $A=(0,\ldots,0)$ and $B=(1,\ldots,1)$ (for example), and for each "$n$-dimensional side" adjacent to $B$ (there are $n$ of them) take the convex hull of $A$ and that side. – Michał Miśkiewicz May 03 '22 at 17:21
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@MichałMiśkiewicz Yes. Or, for each of the $n$ coordinate axes, you get an $n$-pyramid whose altitude is the interval from 0 to 1 along that axis and whose cross sections are $(n-1)$-dimensional cubes given by $0\le x_j \le x_i$, where $x_i$ is the coordinate in the chosen direction and $x_j$ for $1\le j\le n$, $j\ne i$ are the $n-1$ other coordinates. More simply stated, the $i^\text{th}$ pyramid is characterized by the condition that $x_i$ is the coordinate with greatest value. – Will Orrick May 03 '22 at 17:51
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@WillOrrick This description also gives a probabilistic interpretation of the formula $\int_0^1 x^n \textrm{d}x = \frac{1}{n+1}$: https://mathoverflow.net/a/114739/1106 – Steven Gubkin May 04 '22 at 11:53
"[Math] is a deductive art." I would actually go so far as to say it is an inductive art. – Arthur May 05 '22 at 09:26
@Arthur Unfortunately, mathematical induction is actually a form of deductive reasoning. https://en.wikipedia.org/wiki/Inductive_reasoning – Steven Gubkin May 05 '22 at 12:11
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I find it a bit easier to imagine $6$ pyramids of height $a/2$ with bases on the cube faces and the apex at the cube center, but that is certainly a matter of taste. – fedja Jun 01 '22 at 00:47
@fedja I actually teach the argument your way to future educators (since I agree: it is easier to visualize), but it complicates the $\frac{1}{3}$ with an extra factor of $\frac{1}{2}$. If you do not an excellent grasp of multiplication of fractions this can trip you up. – Steven Gubkin Jun 01 '22 at 13:07

score 11 · Answer 2 · answered May 02 '22 at 13:23

One framework of understanding and building towards proof that we teach to mathematics teachers at primary school level is (and translated to English on fly):

Naive empiricism: The pupil tries an example, or maybe several, and checks if the claim holds for them.
Crucial experiment: The pupil tries an example chose with care such that if something should be wrong with a claim, this example would show it. The cruciality is not formally proven, however.
Generic example: Here one takes a representative example and uses it to show an argument that works for any example, but this generality is not formalized (by, for example, using variables).
General argument: An idea of proof, a mathematician would say.
Formal argument: A mathematical proof.

There is some overlap between these, and some of these are also proof strategies when formalized.

I think one to three are directly from

Balacheff, N. (1988): Aspects of proof in pupils’ practice of school mathematics,. En Pimm, D. (ed.), Mathematics, teachers and children (Hodder & Stoughton: Londres), 216-235.

His fourth category might be a bit different than the ones I use here.

The video

Looking at the video, the presenter demonstrates the claim with a single example and invites (claims) it to hold for others, too. So we are at least at the level of naive empiricism.

We might argue that there is nothing special about these particular cylinders (they seem to be of different ratios of height to radius) and it is also unlikely that something whether this ratio is a rational number or not, or some other occult property, would have any effect. So maybe this is a generic example? But the video does not argue this; I am over-interpreting here. But not much.

We did not see this done with a cylinder that has very extreme form, like ratio of height to radius being very small or very large. Neither did we see the exercise done with a great deal of accuracy. Had we seen such, maybe we could say that this amounts to a generic example: the claim would have been demonstrated with a usual, an extremely sharp and extremely dull object, and thus it would have seen likely that it also holds for other shapes. Alas.

Se we certainly saw a sloppy example, and were implied to that this same should hold generally, so we are at least at naive empiricism and maybe, maybe, seeing a generic example.

Thanks for this perspective. I would say it is clearly at the level of "naive empiricism" and not at all "generic example". If they had chosen some particular cone (like a cone with a regular pentagon base) and given an actual argument that its volume was $\frac{1}{3}Bh$ in this particular instance, I think that would qualify as a "generic example". It would have to be an argument which could "clearly generalize" to all other cases. — Steven Gubkin, May 02 '22 at 17:07

score 6 · Answer 3 · answered May 02 '22 at 20:41

Without Calculus, you can still be sort of rigorous. With Euclidean geometry we can find the centroid of a right triangle by finding the point where all the medians intersect. A cardboard triangle will balance on a pin at its centroid. We also know that the centroid cuts each median into 1/3 and 2/3's of its length. (This is where the 1/3 comes in.)

Second, we have the (second) Theorem of Pappus (which takes some sort of calculus to prove, but which would seem quite reasonable to elementary students) which says that he volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F.

A cone is formed by rotating a right angle about one leg. We can calculate its volume using these two theorems.

score 2 · Answer 4 · answered May 05 '22 at 14:47

A possibility may be to show how a cube consists of 6 equivalent pyramids. The picture below shows three of them, and has empty slots in between for the missing three (I felt this would make the picture easier to comprehend). Each of the pyramids has height equal to the edge of the cube, and base that is one half of one of the faces of the cube. The only way the volumes can add up is if we have the correct formula $V=\frac13Ah=\frac13\cdot\frac12a^2\cdot a=\frac16a^3$.

As a bonus you also see the following. Assume that the cube in the picture is the set $[0,1]\times[0,1]\times[0,1]$ in $\Bbb{R}^3$, and the edge shared by all the six parts is the 3D-diagonal from $(0,0,0)$ to $(1,1,1)$. Then the coordinates points $(x,y,z)$ in any of the six parts share the same ordering. For example, all the points in one cone satisfy $x\le y\le z$ et cetera. Obviously there are six possible orderings for the three coordinates. Observe that the three planes $x=y$, $y=z$ and $z=x$ can also be seen in the picture, as all the six cones have a face on two of those planes.

Explaining why volume of cone is a third of cylinder

4 Answers4