A prismatoidMathworldPlanetmath is a polyhedron, possibly not convex, whose vertices all lie in one or the other of two parallel planesMathworldPlanetmath. 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 polygonMathworldPlanetmathPlanetmath formed by cutting the prismatoid by a plane parallelMathworldPlanetmath to the bases halfway between them.

The volume of a prismatoid is given by the prismoidal formula:


where h is the altitude, B1 and B2 are the areas of the bases and M is the area of the midsection.

An alternate formula is :


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 distanceMathworldPlanetmath from B1 to B2.

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 trapezoidsMathworldPlanetmath. However, this condition is unnecessary since it is easily shown to hold.


  • 1 A. Day Bradley, Prismatoid, Prismoid, Generalized Prismoid, The American Math. Monthly, 86, (1979), 486-490.
  • 2 G.B. Halsted, Rational GeometryMathworldPlanetmath: A textbook for the Science of Space. Based on Hilbert’s Foundations, second edition, John Wiley and Sons, New York, 1907
Title prismatoid
Canonical name Prismatoid
Date of creation 2013-03-22 17:12:03
Last modified on 2013-03-22 17:12:03
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 10
Author Mathprof (13753)
Entry type Definition
Classification msc 51-00
Related topic SimpsonsRule
Related topic Volume2
Related topic TruncatedCone
Defines altitude
Defines bases
Defines prismoidal formula