We use the concept of cross-sectional area: , where gives the area of the cross-section created by slicing through the figure at , and h is the height.
Picture a pyramid with its apex at the origin and the center of its base sitting on the point on the x-axis, where is of course its height. Now imagine cutting the pyramid with a vertical sheet parallel to the plane passing through the point for . The cross sectional area is going to be a square. This square forms a "miniature" pyramid that is a scaled down model of the original. So, the ratio of x to s, the side of the cross sectional square, is the same as the ratio of h to b, the base side of the whole pyramid: . Thus, the area of this square is
It is a known fact from the Greeks that the volume of any "pointed" object, that is, a two-dimensional shape as a base that comes to a point has an area that is one third its height times the area of its base.