Some general principles: Suppose and are two similar
figures. Assume that the ratio of two corresponding
one-dimensional lengths is *a*/*b*. Then the ratio of two corresponding
two-dimensional areas is , and the ratio of two corresponding
three-dimensional volumes is .

The volume of a prism with base of area *B* and height *h* is *Bh*
(area of base times height).

The volume of a pyramid with base of
area *A* and height *h* is . You can prove this with
calculus by placing the pyramid with apex at the origin and
with base perpendicular to the
*z*-axis. Then we can calculate the volume by making slices
perpendicular to the *z*-axis. Let *A*(*z*) be the area of the slice
at position *z*. Then , so .
We get the volume by integrating from *z*=0 to *z*=*h*:

Wed Apr 21 08:17:28 EDT 1999