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# prismatoid

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.

# References

- 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

## Mathematics Subject Classification

51-00*no label found*

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