Pentagonal prism

Uniform pentagonal prism
File:Pentagonal prism.png
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}×{}
Wythoff symbol	2 5 | 2
Coxeter diagram	File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 5.pngFile:CDel 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
File:Pentagonal prism vertfig.png Vertex figure 4.4.5

File:Pentagonal prism.stl

3D model of a (uniform) pentagonal prism

In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten 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}×{}. 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

$\frac{h^3}{4}\sqrt{5(5+2\sqrt{5})}\approx 1.72h^3$

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 four dimensions:

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

Related polyhedra

File:Stephanoid5.png

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

External links

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