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I have come back to study geometry a bit and I'm kind of stuck at deriving the volume formula for a cone. I have read the calculus-based derivation and it totally makes sense, but calculus has been around for 200+ years, cones have been around forever.

Intuition leads me to believe that there must be a way for people to logically explain that a cone in 1/3th the volume of a cilinder of the same size before calculus was even a thing. (similarly to the way that the area equation of a circle can be derived from breaking down the circle into infinite triangular slices.)

Is there any logical way to get to a cone's volume equation without calculus? can't it be explained using some geometrical argument? How did civilizations wrap their heads around a cone's volume before calculus?

Rick
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Joaquin Brandan
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  • I don't know if this is a " logical way to get to a cone's volume equation without calculus", but it is definitely a nice ad-hoc way of visualizing this fact. – Rick Jul 08 '20 at 21:00
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    Cf. this question and several answers there – J. W. Tanner Jul 08 '20 at 21:02
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    @J.W.Tanner: Oops, I just noticed the situation I describe is exactly the top answer to that question. – Brian Tung Jul 08 '20 at 21:10
  • There's hardly a way to deal with measuring lengths, areas and volumes without integration, or some version of it. – Allawonder Jul 08 '20 at 21:25
  • For example when you say "the area equation of a circle can be derived from breaking down the circle into infinite triangular slices" that's precisely an idea of integration. I think what you meant was an intuitive explanation, not one devoid of the infinitesimal calculus (if such even exists). – Allawonder Jul 08 '20 at 21:26
  • @Chrystomath I think so, I'm still reading and sorting all the information. – Joaquin Brandan Jul 08 '20 at 21:36
  • @Allawonder I'm looking for a plausible way of logically getting an idea of why that formula is true before newton. they didnt have the "infinite" concept but they did just try to use a ton of triangles to go about it and get an aproximate answer, so while it's not "infinite" they aparently did had the idea of trying to get a good aproximation. One may argue this is indeed "numerical integration", and that's ok. – Joaquin Brandan Jul 08 '20 at 21:41
  • Euclid's proof is basically the answer by Brian Tung. – Chrystomath Jul 08 '20 at 21:48
  • The problem of "infinity" already occurs in finding the area of a circle. – Chrystomath Jul 08 '20 at 21:50
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    @JoaquinBrandan Integration existed before Newton. In fact, much before him. Again, if you want an exact treatment of measurement of figures, you cannot escape infinitesimal methods, no matter how disguised. – Allawonder Jul 08 '20 at 23:13
  • http://www.ams.org/publicoutreach/feature-column/fcarc-archimedes3 – brainjam Jul 09 '20 at 16:57

2 Answers2

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ETA: Based on the comments, I should make it clear upfront that this is not an explanation that is free of calculus. It avoids much of the mechanical manipulations of integral calculus, but the basic notions are in there, though "dressed up" in a way that hopefully conveys some intuition about how the formula comes about.

One possibility is to notice that you can dissect a unit cube into three congruent portions, each of which is a skew pyramid with the same one vertex as the apex, and one of the three opposite squares as the base. Therefore, the volume of those skew pyramids is $1/3$, or equivalently, equal to one third times the height times the area of the base: $V = \frac13Bh$.

Then imagine taking any of the skew pyramids and cutting it into infinitesimal square slices parallel to its base, and then "straightening" it out. That should not change the volume, so we still have $V = \frac13Bh$. If we stretch the pyramid out, we may change $B$ or $h$, but you may convince yourself that we still have $V = \frac13Bh$.

Finally, if we take each square slice and shave off everything except the circle we inscribe inside it, then clearly the remaining area of each slice (and therefore of the base) is reduced in the same proportion as the overall volume, so we still have

$$ V = \frac13Bh $$

Of course, the foregoing is hardly a proof, but it may serve to satisfy intuition, perhaps.

Brian Tung
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  • When you break the pyramids into "infinitesimal square slices," that's still an idea of the calculus. As I've commented under OP above, I doubt that an explanation of the concept of measure can be done without such ideas. – Allawonder Jul 08 '20 at 21:30
  • @Allawonder: I don't disagree, but the impact of that depends on just why OP wants a "non-calculus" answer. If it's just a matter of the mechanics of calculus, I don't think it's a big problem. – Brian Tung Jul 08 '20 at 21:52
  • already the area of a circle must be computed using calculus (or exhaustion) – mau Jul 09 '20 at 07:04
  • @mau: Or something equivalent, yes. However, if one is willing to take the base area as a given (as in the formula $V = \frac13Bh$), that issue is finessed. – Brian Tung Jul 09 '20 at 15:41
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I'm answering my own question in hope that this might prove useful for anyone in the future.

First, this is indeed a duplicate of this question, so I will mark it as such.

Second, as many here pointed out, ancient civilizations knew quite a bit about infinitesimal calculus and numerical integration.

Third, Exodus (337BC) proved the volume of a pyramid using such infinitesimal calculus methods, which can be extended to prove the cylinder/cone relationship as well. This method can be visualized here.

fourth, There are other, more modern methods of deriving the volue formula that still do not require "newton's/lebnitz calculus methods". Mainly using cavallier's principle (1600AC), that can be visualized here or here

Thanks to everyone who answered and commented on this question, I have learned a lot.

Joaquin Brandan
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