Ah yes, but what interesting, related shape has the volume of 3 times that of a cylinder of a certain base and height, I wonder? Perhaps the coefficient here is just coincidentally interesting and there are actually infinitely interesting shapes connected in this manner.
The general result is that any volume consisting of a stack of cross sections whose area goes does as the square of the distance from the base will have a volume of 1/3 the area of the base times the height. The usual special cases are squares (pyramids) and circles (cones) but it works for any shape. It doesn't even have to be convex.
