# Elongated pentagonal cupola

Elongated pentagonal cupola
Rank3
TypeCRF
Notation
Bowers style acronymEpcu
Coxeter diagramoxx5xxx&#xt
Elements
Faces5 triangles, 5+5+5 squares, 1 pentagon, 1 decagon
Edges5+5+5+5+5+10+10
Vertices5+10+10
Vertex figures5 isosceles trapezoids, edge lengths 1, 2, (1+5)/2, 2
10 trapezoids, edge lengths 1, 2, 2, 2
10 isosceles triangles, edge lengths (5+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}}{6}}\approx 10.01825}$
Dihedral angles3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–4 prismatic: 144°
3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }}$
4–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
4–10: 90°
Central density1
Number of external pieces22
Level of complexity18
Related polytopes
ArmyEpcu
RegimentEpcu
DualDecakis order-10 truncated semibisected pentagonal trapezohedron
Abstract & topological properties
Flag count180
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal cupola is one of the 92 Johnson solids (J20). It consists of 5 triangles, 5+5+5 squares, 1 pentagon, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal cupola.

If a second cupola is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobicupola. If the second cupola is rotated 36º instead, the result is the elongated pentagonal gyrobicupola.

## Vertex coordinates

An elongated pentagonal cupola of edge length 1 has the following vertices:

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