My math teacher handed this out today without explaining, and told us to try and solve it. He gave us the formula to solve the volume of a pyramid (1/3 x area of base x height) and told us to explain WHY it was like that. I already figured out how to explain the surface area of the tetrahedron (he told us to figure out the surface area, without the base, and volume for a square based pyramid first, and then just multiply by two), but I'm a bit stuck on the volume part. He figured this one would stump us since, although we understand how to find a triangle's area, (b x h)/2, we wouldn't understand a pyramid. He also said we should follow the way of Eudoxus who claimed: a pyramid can be approximated to a collection of slabs of reducing size. For example, a square-based pyramid is comparable to pile of square based cuboids. By increasing the number of slabs and reducing their heights, the approximation would improve. The theorem is v=(Abase x h)/3, where "h" is the perpendicular height. The proof, or what I think it is, is: Consider any pyramid, perpendicular height "h" and with area of base "A" I believe, you then split the pyramid into n layers. Let n be the unknown number of layers and k be the layer you are measuring. So, by similarity the kth layer will have a base with dimension k/n as a fraction of the original base, right? Then the area of the base would be (k/n)squared x A. The area of the base of each layer will be (1/n)squared x A, (2/n)squared x A,..., (n/n)squared x A. That's about all I have so far. I'm not quite sure how that works out; my friend taught me it, so I don't totally understand it. Like what does it mean "by similarity the kth layer will have a base with dimension k/n as a fraction of the original base"?