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?

  • $\begingroup$ 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. $\endgroup$ – Rick Jul 8 '20 at 21:00
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    $\begingroup$ Cf. this question and several answers there $\endgroup$ – J. W. Tanner Jul 8 '20 at 21:02
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    $\begingroup$ @J.W.Tanner: Oops, I just noticed the situation I describe is exactly the top answer to that question. $\endgroup$ – Brian Tung Jul 8 '20 at 21:10
  • $\begingroup$ There's hardly a way to deal with measuring lengths, areas and volumes without integration, or some version of it. $\endgroup$ – Allawonder Jul 8 '20 at 21:25
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    $\begingroup$ @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. $\endgroup$ – Allawonder Jul 8 '20 at 23:13

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.

  • $\begingroup$ 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. $\endgroup$ – Allawonder Jul 8 '20 at 21:30
  • $\begingroup$ @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. $\endgroup$ – Brian Tung Jul 8 '20 at 21:52
  • $\begingroup$ already the area of a circle must be computed using calculus (or exhaustion) $\endgroup$ – mau Jul 9 '20 at 7:04
  • $\begingroup$ @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. $\endgroup$ – Brian Tung Jul 9 '20 at 15:41

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.


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