Without using calculus, it’s actually pretty mysterious where that 1/3 factor comes from. I found this somewhat shocking—- usually there is some easy (but perhaps clever or elegant) framing whereby the answer just “falls out,” but not in this case!
There is the demonstration analyzing a frustum and then extending its smaller radius until it is the same as the base one (i.e. stretching the frustum until it becomes a cylinder):
This does not use calculus at all but still needs some time to digest, the idea is that there must be a coefficient c that multiplied by the base area of the cone A and by its height gives the volume (V=cAh) and that if we express the volume of the cylynder with the same approach we get V=3cAh.
I love how this document just randomly throws in "e" which mysteriously means "extra" with absolutely zero further explanation or rationale. This is why mathematicians drive me insane. There is zero boilerplate for explaining where these variables come from and why.
a frustum is the base of a cone when you slice off a conical cap, so the remaining height of the cone from which the frustum was sliced is the "extra".
