Pentagonal cupola

Pentagonal cupola
	(3D model); (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Pecu
Coxeter diagram	ox5xx&#x
Elements
Faces	5 triangles, 5 squares, 1 pentagon, 1 decagon
Edges	5+5+5+10
Vertices	5+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	10 scalene triangles, edge lengths 1, √2, √(5+√5)/2
Measures (edge length 1)
Circumradius
Volume
Dihedral angles	3–4:
	4–5:
	3–10:
	4–10:
Height
Central density	1
Related polytopes
Army	Pecu
Regiment	Pecu
Dual	Semibisected pentagonal trapezohedron
Conjugate	Retrograde pentagrammic cupola
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

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[edit | edit source]

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:

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

A pentagonal cupola has the following Coxeter diagrams:

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.