# Pentagonal cupola

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Pentagonal cupola Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymPecu
Coxeter diagramox5xx&#x
Elements
Faces5 triangles, 5 squares, 1 pentagon, 1 decagon
Edges5+5+5+10
Vertices5+10
Vertex figures5 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$\frac{\sqrt{11+4\sqrt5}}{2} ≈ 2.23295$ Volume$\frac{5+4\sqrt5}{6} ≈ 2.32404$ Dihedral angles3–4: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$ 4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ 3–10: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°$ 4–10: $\arccos\left(\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 31.71747°$ Height$\sqrt{\frac{5-\sqrt{5}}{10}}\approx 0.52573$ Central density1
Related polytopes
ArmyPecu
RegimentPecu
DualSemibisected pentagonal trapezohedron
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

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:

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