Pentagonal cupola

The pentagonal cupola is one of the 92 Johnson solids (J5). It consists of 5 triangles, 5 squares, 1 pentagon, and 1 decagon. It is a cupola based on the pentagon, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the square cupola.

It can be obtained as a segment of the small rhombicosidodecahedron.

Vertex coordinates
A pentagonal cupola of edge length 1 has vertices given by the following coordinates:


 * $$\left(\pm\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$\left(\pm\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{10}}\right),$$
 * $$\left(\pm\frac12,\,\pm\frac{\sqrt{5+2\sqrt5}}{2},\,0\right),$$
 * $$\left(\pm\frac{3+\sqrt5}{4},\,\pm\sqrt{\frac{5+\sqrt5}{8}},\,0\right),$$
 * $$\left(\pm\frac{1+\sqrt5}{2},\,0,\,0\right).$$

These can be obtained from placing a pentagon and decagon in parallel planes.

Alternatively, coordinates can be obtained as a subset of vertices of the small rhombicosidodecahedron:


 * $$\left(\pm\frac12,\,\pm\frac12,\,\frac{2+\sqrt5}{2}\right),$$
 * $$\left(0,\,\pm\frac{3+\sqrt5}{4},\,\frac{5+\sqrt5}{4}\right),$$
 * $$\left(\frac{5+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{2}\right),$$
 * $$\left(\frac{5+\sqrt5}{4},\,0,\,\frac{3+\sqrt5}{4}\right),$$
 * $$\left(\frac{1+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{3+\sqrt5}{4}\right),$$
 * $$\left(\frac{1+\sqrt5}{2},\,\pm\frac{3+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(\frac{2+\sqrt5}{2},\,\pm\frac12,\,\frac12\right).$$

Representations
A pentagonal cupola has the following Coxeter diagrams:


 * ox5xx&#x
 * so10ox&#x

Related polyhedra
Two pentagonal cupolas can be attached at their decagonal bases in the same orientation to form a pentagonal orthobicupola. If the second cupola is rotated by 36º the result is the pentagonal gyrobicupola. If a pentagonal rotunda is attached, the result is either a pentagonal orthocupolarotunda (if the base pentagons are in the same orientation) or a pentagonal gyrocupolarotunda (if the base pentagons are rotated 36º).

A decagonal prism can be attached to the pentagonal cupola's decagonal base to form the elongated pentagonal cupola. If a decagonal antiprism is attached instead, the result is the gyroelongated pentagonal cupola.

The pentagonal cupola is the pentagon-first cap of the small rhombicosidodecahedron. Gyrating or removing such caps results in Johnson Solids 72 to 83. The pentagonal cupola can also be augmented onto a truncated dodecahedron, producing the augmented, parabiaugmented, metabiaugmented and triaugmented truncated dodecahedra.