The volume of a pyramid is 1/3 the base area times the height.
This works whatever the shape of the base is, so it works for a cone too. The key idea is to slice the pyramid into lots of slices. The slices are all the same shape. The ones near the base are the largest, and the slices get smaller and smaller as you move towards the apex. Would someone who know this please take over the explanation?
I've started a page about this on the wiki, and I'd like someone to step in and help complete it.
--James.