Pentagonal prism

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.

Uniform pentagonal prism
TypePrismatic uniform polyhedron
ElementsF = 7, E = 15
V = 10 (χ = 2)
Faces by sides5{4}+2{5}
Schläfli symbolt{2,5} or {5}×{}
Wythoff symbol2
Coxeter diagram
Symmetry groupD5h, [5,2], (*522), order 20
Rotation groupD5, [5,2]+, (522), order 10
ReferencesU76(c)
DualPentagonal dipyramid
Propertiesconvex

Vertex figure
4.4.5
3D model of a (uniform) pentagonal prism

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 D5h 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

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
The pentagonal stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism.
  • Weisstein, Eric W. "Pentagonal prism". MathWorld.
  • Pentagonal Prism Polyhedron Model -- works in your web browser
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