Pentagonal prism

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Uniform Pentagonal prism
Pentagonal prism
Type Prismatic uniform polyhedron
Elements F = 7, E = 15
V = 10 (χ = 2)
Faces by sides 5{4}+2{5}
Schläfli symbol t{2,5} or {5}x{}
Wythoff symbol 2 5 | 2
Coxeter-Dynkin CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 5.pngCDel node.png
Symmetry group D5h, [5,2], (*522), order 20
Rotation group D5, [5,2]+, (522), order 10
References U76(c)
Dual Pentagonal dipyramid
Properties convex
Pentagonal prism
Vertex figure
4.4.5

In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.

As a semiregular (or uniform) polyhedron[edit]

If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}x{}. The dual of a pentagonal prism is a pentagonal bipyramid.

The symmetry group of a right pentagonal prism is D5h of order 20. The rotation group is D5 of order 10.

Volume[edit]

The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism with edges h the formula is

\frac{h^3}{4}\sqrt{5(5 + 2\sqrt{5})}

Use[edit]

Nonuniform pentagonal prisms called pentaprisms are also used in optics to rotate an image through a right angle without changing its chirality.

In 4-polytopes[edit]

It exists as cells of four nonprismatic uniform 4-polytopes in 4 dimensions:

cantellated 600-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
cantitruncated 600-cell
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
runcinated 600-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
runcitruncated 600-cell
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
600-cell t02 H3.svg 120-cell t123 H3.png 120-cell t03 H3.png 120-cell t023 H3.png

Related polyhedra[edit]

The pentagonal stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism.
Family of uniform prisms
Symmetry 3 4 5 6 7 8 9 10 11 12
[2n,2]
[n,2]
[2n,2+]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 10.pngCDel node h.pngCDel 2.pngCDel node h.png
CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node 1.pngCDel 12.pngCDel node h.pngCDel 2.pngCDel node h.png
Image Triangular prism.png Tetragonal prism.png
Uniform polyhedron 222-t012.png
Cube rotorotational symmetry.png
Pentagonal prism.png Hexagonal prism.png
Truncated triangle prism.png
Cantic snub hexagonal hosohedron.png
Prism 7.png Octagonal prism.png
Truncated square prism.png
Cantic snub octagonal hosohedron.png
Prism 9.png Decagonal prism.png Hendecagonal prism.png Dodecagonal prism.png
As spherical polyhedra
Image Spherical triangular prism.png Spherical square prism.png
Spherical square prism2.png
Spherical pentagonal prism.png Spherical hexagonal prism.png
Spherical hexagonal prism2.png
Spherical heptagonal prism.png Spherical octagonal prism.png
Spherical octagonal prism2.png
Spherical decagonal prism.png
Spherical decagonal prism2.png

External links[edit]