right|thumb|240px|Prismatoid with parallel faces and , midway cross-section , and height .

In geometry, a prismatoid is a convex polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a prismoid.

Volume

If the areas of the two parallel faces are and , the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is , and the height (the distance between the two parallel faces) is , then the volume of the prismatoid is given by

<math display="block">V = \frac{h(A_1 + 4A_2 + A_3)}{6}.</math>

This formula follows immediately by integrating the area parallel to the two planes of vertices by Simpson's rule, since that rule is exact for integration of polynomials of degree up to 3, and in this case the area is at most a quadratic function in the height.

Prismatoid families

{| class=wikitable

!Pyramids

!Wedges

!Parallelepipeds

!colspan=1|Prisms

!colspan=3|Antiprisms

!Cupolae

!Frusta

|-

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Families of prismatoids include:

  • Pyramids, in which one plane contains only a single point;
  • Wedges, in which one plane contains only two points;
  • Prisms, whose polygons in each plane are congruent and joined by rectangles or parallelograms;

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