The Prismatoid and the Prismoidal Formula

The General Prismatoid

A general prismatoid is a solid such that the area of any section, say A_y, parallel to and distant 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
A₁ and A₂ = areas of parallel bases
A_m = area of the section midway between A₁ and A₂
L = perpendicular distance between A₁ and A₂
 

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