{{Short description|Polyhedron with all vertices in two parallel planes}} right|thumb|240px|Prismatoid with parallel faces {{math|''A''{{sub|1}}}} and {{math|''A''{{sub|3}}}}, midway cross-section {{math|''A''{{sub|2}}}}, and height {{mvar|h}}.
In geometry, a '''prismatoid''' is a convex polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles.{{r|prismatoid}} If both planes have the same number of vertices, and the lateral faces are either parallelograms or trapezoids, it is called a '''prismoid'''.{{r|an}}
==Volume== If the areas of the two parallel faces are {{math|''A''{{sub|1}}}} and {{math|''A''{{sub|3}}}}, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is {{math|''A''{{sub|2}}}}, and the height (the distance between the two parallel faces) is {{mvar|h}}, then the volume of the prismatoid is given by{{r|meserve}} <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 |- |80px |100px |80px |80px |80px |80px |80px |80px |80px |}
Families of prismatoids include:
*Pyramids, in which one plane contains only a single point;<ref name="grunbaum">{{cite journal | last = Grünbaum | first = Branko | year = 1997 | title = Isogonal Prismatoids | journal = Discrete & Computational Geometry | volume = 18 | pages = 13–52 | doi = 10.1007/PL00009307 }}.</ref> *Wedges, in which one plane contains only two points; *Prisms, whose polygons in each plane are congruent and joined by rectangles or parallelograms;<ref name="grunbaum"/> *Antiprisms, whose polygons in each plane are congruent and joined by an alternating strip of triangles;{{sfnp|Alsina|Nelsen|2015|p=[https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA87 87]}} *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: *# 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== thumb|215x215px|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.
==References== <references> <ref name="an">{{cite book | last1 = Alsina | first1 = Claudi | last2 = Nelsen | first2 = Roger B. | year = 2015 | title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century | volume = 50 | publisher = Mathematical Association of America | url = https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA85 | page = 85 | isbn = 978-1-61444-216-5 }}</ref>
<ref name="meserve">{{cite journal | last1 = Meserve | first1 = B. E. | last2 = Pingry | first2 = R. E. | title = Some Notes on the Prismoidal Formula | journal = The Mathematics Teacher | volume = 45 | issue = 4 | year = 1952 | pages = 257–263 | doi = 10.5951/MT.45.4.0257 | jstor = 27954012}}</ref>
<ref name="prismatoid">{{cite book | last1 = Kern | first1 = William F. | last2 = Bland | first2 = James R. | title = Solid Mensuration with proofs | url = https://books.google.com/books?id=Y6cAAAAAMAAJ | year = 1938 | page = 75}}</ref> </references>
==External links== *{{MathWorld|urlname=Prismatoid|title=Prismatoid}}
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Category:Prismatoid polyhedra