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.

Vertex coordinates
A pentagonal prism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,±\frac12\right),$$
 * $$\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right).$$

Representations
A pentagonal prism has the following Coxeter diagrams:

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

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

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:


 * Rhombidodecahedron (6)
 * Disrhombidodecahedron (12)

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.