Pentagonal prism

Uniform 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 diagram
Symmetry group	D5h, [5,2], (*522), order 20
Rotation group	D5, [5,2]^+, (522), order 10
References	U76(c)
Dual	Pentagonal dipyramid
Properties	convex
Vertex figure 4.4.5

In geometry, the pentagonal prism 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

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 D_5h of order 20. The rotation group is D₅ of order 10.

Volume

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

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

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

cantellated 600-cell	cantitruncated 600-cell	runcinated 600-cell	runcitruncated 600-cell

Related polyhedra

The pentagonal stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism.

Family of uniform prisms

Polyhedron
Coxeter
Tiling
Config.	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4

External links

• Weisstein, Eric W. "Pentagonal prism". MathWorld.

• Pentagonal Prism Polyhedron Model -- works in your web browser