prismatoid

A prismatoid is a polyhedron, possibly not convex, whose vertices all lie in one or the otherof two parallel planes.The perpendicular distance between the two planes is called the altitudeof the prismatoid.The faces that lie in the parallel planes are called the basesof the prismatoid.The midsection is the polygon formed by cutting the prismatoid bya plane parallel to the bases halfway between them.

The volume of a prismatoid is given by the prismoidal formula:

$$V=\frac{1}{6}h(B_1+B_2+4M)$$

where $h$ is the altitude, $B_1$ and $B_2$ are the areas of the bases and $M$is the area of the midsection.

An alternate formula is :

$$V=\frac{1}{4}h(B_1+3S)$$

where $S$ is the area of the polygon that is formed by cutting the prismatoidby a plane parallel to the bases but 2/3 of the distance from $B_1$ to $B_2$.

A proof of the prismoidal formula for the case wherethe prismatoid is convex is in [1]. It is also proved in [2] for any prismatoid.The alternate formula is proved in [2].

Some authors impose the condition that the lateral faces must be trianglesor trapezoids. However, this condition is unnecessary since it is easily shownto hold.