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Does anyone have a good mnemonic for the volume of cone formula: $$V=\frac{1}{3}\pi r^2 h?$$

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    $\begingroup$ Not a mnemonic, but still rather unrigorous, the way I remember it is you have to have $\pi r^2$ because of the circle (and thinking of Cavalieri's principle), and hence you have to have $h$ (to be dimensionally correct), and then you use $\frac{1}{3}$ because we're in dimension 3 (and in any event, the fraction has to be less than $\frac{1}{2},$ the value you'd use for a triangle, because volumes approach zero faster than areas when linear dimensions approach zero). $\endgroup$ – Dave L Renfro Feb 15 at 11:04
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    $\begingroup$ These are both answers to the question. Why not post them as answers? $\endgroup$ – Chris Cunningham Feb 15 at 14:10
  • $\begingroup$ Not a mnemonic, but I just think of the cone as being cut out of a cylinder. It would be half if in two dimensions. Think about cutting a line down the middle. But in three dimensions, you spin it and get to a third. Once they have calculus you can actually prove this. But for now, just go with hokey spinning it. After all you are just looking for a memory trick for a formula as given. $\endgroup$ – guest Feb 15 at 14:29
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    $\begingroup$ Asking for a mnemonic presupposes that it's a good idea to force students to memorize this sort of thing. It isn't. $\endgroup$ – Ben Crowell Feb 15 at 14:45
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    $\begingroup$ @BenCrowell: Presupposing what someone asking a question presupposes is an even worse idea. (I may be a student or teacher in an education system that forces students to memorize this formula. It may be beyond my power to do anything about the education system, but in my power to figure out how best students can nonetheless navigate it. Arrogant moralizing about what I presuppose is hardly helpful.) $\endgroup$ – user20311 Feb 17 at 1:22
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The comments give great suggestions for how to understand and remember the formula, but since you asked for a mnemonic specifically, how's this?

Henry won third in the creative desserts contest because his pie are squared.

$h\cdot (1/3) \pi r^2$

Edit: someone "fixed" this by changing "are" to "is." The mnemonic falls apart if you do that, so please leave it as-is!

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I'm not sure if I would class this as a mnemonic, except in the most basic definition of "a way to remember":

A cone is basically a three-dimensional triangle. Triangles have an area of $\frac12bh$; but triangles are two-dimensional, so it makes sense that a three-dimensional version would use $3$ instead of $2$, yielding $\frac13bh$. The "base" of a cone is a circle, so $b$ should be the area of that circle: $\pi r^2$. Putting it together, we have $\frac13\pi r^2h$. (This approach has the advantage of working with generalized cones, too, so you can use it to remember that the volume of a square-based pyramid is $\frac13s^2h$, for example.)

Or, building it another way: Because we're looking at three-dimensional volume, we know we should have three lengths multiplied together. On a cone, both the width and the height should matter, so both $r$ and $h$ should appear in the formula. Only one "direction" of a cone is the height; the radius appears in two different directions. So the volume should be a constant times $r^2h$. It's round, so it should have $\pi$ somewhere. But $\pi r^2h$ is the volume of a cylinder, which occupies more space than a cone does, so we need to make it smaller. Because it's $3$-dimensional, we divide by $3$, yielding $\frac13\pi r^2h$.

Of course both of these are kind of nonsense (especially the bit where we figure out the $\frac13$). But if you're just looking for a way to remember (or recover the formula when you forget), it doesn't matter if it's a little bit nonsense.

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Another mnemonic attempt (despite not having much sense):

The cones in my pie are squared, because their age is over three.

Getting together, we have that the volume of a cone is $\pi r^2\cdot \dfrac h3$.

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