A prismatoid is a polyhedron, possibly not convex, whose vertices all lie in one or the other of two parallel planes. The perpendicular distance between the two planes is called the altitude of the prismatoid. The faces that lie in the parallel planes are called the bases of the prismatoid. The midsection is the polygon formed by cutting the prismatoid by a 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 prismatoid by 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 where the 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 triangles or trapezoids. However, this condition is unnecessary since it is easily shown to hold.

Bibliography

1

A. Day Bradley, Prismatoid, Prismoid, Generalized Prismoid, The American Math. Monthly, 86, (1979), 486-490.

2

G.B. Halsted, Rational Geometry: A textbook for the Science of Space. Based on Hilbert's Foundations, second edition, John Wiley and Sons, New York, 1907