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Prismatoid

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From Wikipedia, the free encyclopedia

Polyhedron with all vertices in two parallel planes

Prismatoid with parallel facesA₁ andA₃, midway cross-sectionA₂, and heighth

Ingeometry, aprismatoid is apolyhedron whosevertices all lie in two parallelplanes. Its lateral faces can betrapezoids ortriangles.^[1] If both planes have the same number of vertices, and the lateral faces are eitherparallelograms or trapezoids, it is called aprismoid.^[2]

Volume

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If the areas of the two parallel faces areA₁ andA₃, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces isA₂, and the height (the distance between the two parallel faces) ish, then thevolume of the prismatoid is given by^[3] $V={\frac {h(A_{1}+4A_{2}+A_{3})}{6}}.$ This formula follows immediately byintegrating the area parallel to the two planes of vertices bySimpson's rule, since that rule is exact for integration ofpolynomials of degree up to 3, and in this case the area is at most aquadratic function in the height.

Prismatoid families

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Pyramids	Wedges	Parallelepipeds	Prisms	Antiprisms			Cupolae	Frusta

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;^[4]
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 bytruncation of a pyramid or a cone;
Quadrilateral-facedhexahedral prismatoids:
1. Parallelepipeds – sixparallelogram faces
2. Rhombohedrons – sixrhombus faces
3. Trigonal trapezohedra – six congruent rhombus faces
4. Cuboids – six rectangular faces
5. Quadrilateral frusta – anapex-truncated square pyramid
6. Cube – six square faces

Higher dimensions

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In general, apolytope is prismatoidal if its vertices exist in twohyperplanes. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides.