In geometry, a prismatoid is a 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.
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Area
If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height (the distance between the two parallel faces) is h, then the volume of the prismatoid is given by
Prismatoid families
Families of prismatoids include:
- Parallelepipeds – six parallelogram faces
- Rhombohedrons – six rhombus faces
- Trigonal trapezohedra – six congruent rhombus faces
- Cuboids – six rectangular faces
- Quadrilateral frusta – an apex-truncated square pyramid
- Cube – six square faces
Higher dimensions
In general, a polytope is prismatoidal if its vertices exist in two hyperplanes. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides.
A tetrahedral-cuboctahedral cupola.