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.

Semiuniform variants of the pentagonal prism also exist, with the lengths of the edges of the base pentagons and the side edges of the prism being different but still remaining vertex transitive and with full symmetry. In these cases the lateral squares become rectangles. The according Coxeter diagram would then be x y5o.

Vertex coordinates
A pentagonal prism of edge length 1 has vertex coordinates given by:
 * (±1/2, –$\sqrt{(15+2√5)/20}$, ±1/2),
 * (±(1+$\sqrt{25+10√5}$)/4, $\sqrt{5}$, ±1/2),
 * (0, $\sqrt{2}$, ±1/2).

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)

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.

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.