The Slant of a Cone

A cone's height, its radius and its slant height make a right triangle. The height meets the radius at the centre of the base at a right angle, and the slant height is the hypotenuse.

With radius 3 and height 4, Pythagoras gives ${s}^{2} = {3}^{2}+{4}^{2} = 25$, so the slant height is 5. The rule is Slant height: ${s}^{2} = {r}^{2}+{h}^{2}$. Unlike the pyramid there is no half: the radius already runs from the centre to the edge.

The rule runs backwards too. Knowing the slant height and the radius gives the height, which is what the volume needs.