Editing Prismatoid

{{Short description|Polyhedron with all vertices in two parallel planes}}
[[File:Prismatoid (parameters h,A₁,A₂,A₃).svg|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 [[polyhedron]] whose [[vertex (geometry)|vertices]] all lie in two parallel [[Plane (geometry)|planes]]. Its lateral faces can be [[trapezoid]]s or [[triangle]]s.{{r|prismatoid}} If both planes have the same number of vertices, and the lateral faces are either [[parallelogram]]s 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 [[integral|integrating]] the area parallel to the two planes of vertices by [[Simpson's rule]], since that rule is exact for integration of [[polynomial]]s of degree up to 3, and in this case the area is at most a [[quadratic function]] in the height.

==Prismatoid families==
{| class=wikitable
![[Pyramid (geometry)|Pyramids]]
![[Wedge (geometry)|Wedges]]
![[Parallelepiped]]s
!colspan=1|[[Prism (geometry)|Prisms]]
!colspan=3|[[Antiprism]]s
![[cupola (geometry)|Cupolae]]
![[Frustum|Frusta]]
|-
|[[File:Pentagonal pyramid.png|80px]]
|[[File:Geometric wedge.png|100px]]
|[[File:Parallelepiped 2013-11-29.svg|80px]]
|[[File:Pentagonal prism.png|80px]]
|[[File:Square antiprism.png|80px]]
|[[File:Pentagonal_antiprism.png|80px]]
|[[File:Pentagrammic crossed antiprism.png|80px]]
|[[File:Pentagonal cupola.png|80px]]
|[[File:Pentagonal frustum.svg|80px]]
|}

Families of prismatoids include:

*[[Pyramid (geometry)|Pyramids]], in which one plane contains only a single point;
*[[Wedge (geometry)|Wedges]], in which one plane contains only two points;
*[[Prism (geometry)|Prisms]], whose polygons in each plane are congruent and joined by rectangles or parallelograms;
*[[Antiprism]]s, 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 antiprism]]s;
*[[Cupola (geometry)|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;
*[[Frustum|Frusta]] obtained by [[truncation (geometry)|truncation]] of a pyramid or a cone;
*[[Quadrilateral]]-faced [[hexahedron|hexahedral]] prismatoids:
*# [[Parallelepiped]]s – six [[parallelogram]] faces
*# [[Rhombohedron]]s – six [[rhombus]] faces
*# [[Trigonal trapezohedron|Trigonal trapezohedra]] – six congruent rhombus faces
*# [[Cuboid]]s – six rectangular faces
*# [[frustum|Quadrilateral frusta]] – an [[apex (geometry)|apex]]-[[truncation (geometry)|truncated]] [[square pyramid]]
*# [[Cube]] – six square faces

==Higher dimensions==
[[File:4D_Tetrahedral_Cupola-perspective-cuboctahedron-first.png|thumb|215x215px|A tetrahedral-cuboctahedral cupola.]]
In general, a [[polytope]] is prismatoidal if its vertices exist in two [[hyperplane]]s. For example, in four dimensions, two polyhedra can be placed in two parallel 3-spaces, and connected with polyhedral sides.

==References==
{{reflist|refs=
<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>

}}

==External links==
*{{MathWorld|urlname=Prismatoid|title=Prismatoid}}

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[[Category:Prismatoid polyhedra| ]]


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