Pentagonal prism

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.

Pentagonal prism
Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymPip
Coxeter diagramx x5o ()
Elements
Faces5 squares, 2 pentagons
Edges5+10
Vertices10
Vertex figureIsosceles triangle, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{15+2\sqrt5}{20}} ≈ 0.98672}$
Volume${\displaystyle \frac{\sqrt{25+10\sqrt5}}{4} ≈ 1.72046}$
Dihedral angles4–4: 108°
4–5: 90°
Height1
Central density1
Number of pieces7
Level of complexity3
Related polytopes
ArmyPip
RegimentPip
DualPentagonal tegum
ConjugatePentagrammic prism
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
ConvexYes
NatureTame
Discovered by{{{discoverer}}}

Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12\right),}$
• ${\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right).}$

Representations

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

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 ${\displaystyle \sqrt{a^2\frac{5+\sqrt5}{10}+\frac{b^2}{4}}}$  and its volume is given by ${\displaystyle \frac{\sqrt{25+10\sqrt5}}{4}a^2b}$ .

Variations

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

Related polyhedra

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

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.