TITLE: Volume of Pyramid
QUESTION [10 upvotes]: To calculate the volume of a pyramid (not tetrahedron!) you've to use the formula $\frac{1}{3}B\cdot H,$ where $B$ is the area of the base and $H$ is the height.
My question is: why 1/3? Is a pyramid one-third of a cuboid?

REPLY [9 votes]: Consider a triangular prism. It can be divided into three equivalent pyramids,  and so the volume of a triangular pyramid is $1/3$ of the volume of a triangular prism with the same base and height (see figure on the left). A pyramid whose base has $n$ sides may be divided into $n-2$ tetrahedrons. It follows that its volume is $1/3$ of the volume of a prism with the same base and height (see figure on the right).

Explanation concerning the sketch on the left in response to Byron Schmuland's comment. Removing the front pyramid we are left with two equivalent pyramids (one on the left and the other on the right) whose height is the distance from the upper front corner to the prism back face and whose bases are the two halves of the back face. On the other hand the front pyramid is equivalent to the right pyramid (equal bases and equal heights). 
Reference: F. G. - M., §496 and §497 of Cours de Géométrie Élémentaire, 1917.