Prismatoid

File:Prismatoid (parameters h,A₁,A₂,A₃).svg

Prismatoid with parallel faces A₁ and A₃, midway cross-section A₂, and height Template:Mvar.

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

Volume

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 Template:Mvar, then the volume of the prismatoid is given byTemplate:R/superscript $$V=\frac{h(A_1+4A_2+A_3)}{6}.$$ 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

Pyramids	Wedges	Parallelepipeds	Prisms	Antiprisms			Cupolae	Frusta
File:Pentagonal pyramid.png	File:Geometric wedge.png	File:Parallelepiped 2013-11-29.svg	File:Pentagonal prism.png	File:Square antiprism.png	File:Pentagonal antiprism.png	File:Pentagrammic crossed antiprism.png	File:Pentagonal cupola.png	File:Pentagonal frustum.svg

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;
• Antiprisms, whose polygons in each plane are congruent and joined by an alternating strip of triangles;Template:Sfnp
• Star antiprisms;
• Cupolae, in which the polygon in one plane contains twice as many points as the other and is joined to it by alternating triangles and rectangles;
• Frusta obtained by truncation of a pyramid or a cone;
• Quadrilateral-faced hexahedral prismatoids:
  1. Parallelepipeds – six parallelogram faces
  2. Rhombohedrons – six rhombus faces
  3. Trigonal trapezohedra – six congruent rhombus faces
  4. Cuboids – six rectangular faces
  5. Quadrilateral frusta – an apex-truncated square pyramid
  6. Cube – six square faces

Higher dimensions

File:4D Tetrahedral Cupola-perspective-cuboctahedron-first.png

A tetrahedral-cuboctahedral cupola.

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.