A few days ago, we pulled on a historical thread and wound up browsing the early volumes of The Mathematical Gazette. Doing so, we stumbled across a “mathematical note” from 1896 by Alfred Lodge, the first president of the Mathematical Association. Lodge’s note provides a simple derivation for the volume of a cone. Such arguments don’t vary all that much but, however we missed it, we’d never seen the derivation in the very elegant form presented by Lodge. Here is Lodge’s argument, slightly reworded.
Imagine a cone of height H, and imagine slicing the cone at some level, to give a new cone of height h. Whatever h, any such cone is similar, and so the volumes will scale as . That is, for all such cones we have
and our problem is to find , the constant of proportionality. To do so, think of close to . Then the difference in the cones is approximately a cylinder of height , and with the same base as that of our original cone. So, calculating the volumes,
Dividing by and factoring the difference of two cubes, we then have
Letting , we find
Plugging back in and setting gives the desired formula:
The reader is invited to ponder other shapes and other dimensions.