Pentagonal cupola

The pentagonal cupola, or pecu, 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:


 * (±1/2, –$\sqrt{2|}$, $\sqrt{5}$),
 * (±(1+$\sqrt{2}$)/4, $\sqrt{2}$, $\sqrt{(5+√5)/2}$),
 * (0, $\sqrt{11+4√5}$, $\sqrt{(5–√5)/10}$),
 * (±1/2, ±$\sqrt{5}$/2, 0),
 * (±(3+$\sqrt{3}$)/4, ±$\sqrt{15}$, 0),
 * (±(1+$\sqrt{(5+√5)/10}$)/2, 0, 0).

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:


 * (±1/2, ±1/2, (2+$\sqrt{(5+2√5)/15}$)/2),
 * (0, ±(3+$\sqrt{(5+√5)/10}$)/4, (5+$\sqrt{(5+2√5)/20}$)/4),
 * ((5+$\sqrt{(5–√5)/10}$)/4, ±(1+$\sqrt{5}$)/4, (1+$\sqrt{(5+√5)/40}$)/2),
 * ((5+$\sqrt{(5–√5)/10}$)/4, 0, (3+$\sqrt{(5+√5)/10}$)/4),
 * ((1+$\sqrt{(5–√5)/10}$)/4, ±(1+$\sqrt{(5+2√5)}$)/2, ±(3+$\sqrt{5}$)/4),
 * ((1+$\sqrt{(5+√5)/8}$)/2, ±(3+$\sqrt{5}$)/4, (1+$\sqrt{5}$)/4),
 * ((2+$\sqrt{5}$)/2, ±1/2, 1/2).

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.