Pentagonal prism

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Uniform Pentagonal prism
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}x{}
Wythoff symbol 2 5 | 2
Coxeter diagram 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 vertfig.png
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

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.

External links[edit]

  • Weisstein, Eric W. "Pentagonal prism". MathWorld.
  • Pentagonal Prism Polyhedron Model -- works in your web browser