Pentagonal prism

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Pentagonal prism
Bowers style acronymPip
Coxeter diagramx x5o ()
Conway notationP5
Faces5 squares, 2 pentagons
Vertex figureIsosceles triangle, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Dihedral angles4–4: 108°
 4–5: 90°
Central density1
Number of external pieces7
Level of complexity3
Related polytopes
DualPentagonal tegum
ConjugatePentagrammic prism
Abstract & topological properties
Flag count60
Euler characteristic2
SymmetryH2×A1, order 20

The pentagonal prism, or pip, is a prismatic uniform polyhedron. It consists of 2 pentagons and 5 squares. Each vertex joins one pentagon and two squares. As the name suggests, it is a prism based on a pentagon.

Vertex coordinates[edit | edit source]

A pentagonal prism of edge length 1 has vertex coordinates given by:

  • ,
  • ,
  • .

Representations[edit | edit source]

A pentagonal prism has the following Coxeter diagrams:

  • x x5o () (full symmetry)
  • xx5oo&#x (seen as frustum)
  • xxx ofx&#xt (A1×A1 symmetry, side)

Semi-uniform variant[edit | edit source]

The pentagonal prism has a semi-uniform variant of the form x y5o that maintains its full symmetry. This variant uses rectangles as its sides.

With base edges of length a and side edges of length b, its circumradius is given by and its volume is given by .

Variations[edit | edit source]

The main variation of a pentagonal prism, with bases of different sizes, is the pentagonal frustum.

Related polyhedra[edit | edit source]

A pentagonal pyramid can be attached to a base of the pentagonal prism to form the elongated pentagonal pyramid. if a second pentagonal pyramid is attached to the opposite base the result is the elongated pentagonal bipyramid.

It is also possible to augment square faces of the pentagonal prism with square pyramids. If one square is augmented the result is the augmented pentagonal prism. If a second non-adjacent square is also augmented the result is the biaugmented pentagonal prism.

Two non-prismatic uniform polyhedron compounds are composed of pentagonal prisms:

There are an infinite amount of prismatic uniform compounds that are the prisms of compounds of pentagons.

In vertex figures[edit | edit source]

The pentagonal prism appears as the vertex figure of the uniform rectified hexacosichoron. This vertex figure has an edge length of 1. Three other uniform polychora - the rectified great hecatonicosachoron, rectified grand stellated hecatonicosachoron, and rectified great faceted hexacosichoron - have semi-uniform pentagonal prisms for vertex figures.

External links[edit | edit source]