Pentagonal prism


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}×{}
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

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

3D model of a (uniform) pentagonal prism

As a semiregular (or uniform) polyhedronEdit

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.


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



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

In 4-polytopesEdit

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 polyhedraEdit

The pentagonal stephanoid has pentagonal dihedral symmetry and has the same vertices as the uniform pentagonal prism.
Family of uniform n-gonal prisms
Prism name Digonal prism (Trigonal)
Triangular prism
Square prism
Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism Enneagonal prism Decagonal prism Hendecagonal prism Dodecagonal prism ... Apeirogonal prism
Polyhedron image                       ...
Spherical tiling image                 Plane tiling image  
Vertex config. 2.4.4 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 ... ∞.4.4
Coxeter diagram                                                                   ...      

External linksEdit

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