# Pentagonal cupola

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Pentagonal cupola
Rank3
TypeCRF
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${\displaystyle {\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295}$
Volume${\displaystyle {\frac {5+4{\sqrt {5}}}{6}}\approx 2.32404}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
3–10: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
4–10: ${\displaystyle \arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Central density1
Number of external pieces12
Level of complexity10
Related polytopes
ArmyPecu
RegimentPecu
DualSemibisected pentagonal trapezohedron
Abstract & topological properties
Flag count100
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:

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

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:

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

## 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.