# The Prismatoid and the Prismoidal Formula

## The General Prismatoid

A * general prismatoid* is a solid such that the area of any section, say

*A*, parallel to and distant

_{y}*y*from a fixed plane can be expressed as a polynomial of

*y*of degree not higher than the third.

A solid is a general prismatoid if

Where *a*, *b*, *c*, and *d* are arbitrary constants which may be positive, negative, or zero.

Prisms and cylinders in general are prismatoids in which *A _{y}* is constants. Meaning, the value of

*b*,

*c*, and

*d*are zero.

Cones and pyramids including their frustums are first degree prismatoids, meaning, the value of *c* and *d* are zero.

Spheres and segment of spheres are examples of second degree prismatoids.

## The Prismatoid

A * prismatoid* is a polyhedron having bases containing all its vertices in two parallel planes. The lateral faces are triangles or trapezoids or parallelograms with one side lying on one of its two bases.

Cubes, cuboids, prisms, pyramids, and pyramidal frustums are examples of basic prismatoids. Prismatoids having vertices equal in numbers in both bases are called * Prismoids*.

## The Prismoidal Formula

The volume of prismatoid is given by this formula:

Where

*A*_{1} and *A*_{2} = areas of parallel bases

*A*_{m} = area of the section midway between *A*_{1} and *A*_{2}

*L* = perpendicular distance between *A*_{1} and *A*_{2}

**Note:** A solid in which all sections parallel to a certain base are similar figures, is a prismatoid.

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