# Pentagonal orthobicupola

Pentagonal orthobicupola
Rank3
TypeCRF
Notation
Bowers style acronymPobcu
Coxeter diagramxxx5oxo&#xt
Elements
Faces
Edges5+5+10+20
Vertices10+10
Vertex figures10 isosceles trapezoids, edge lengths 1, 2, (1+5}/2, 2
10 kites, edge lengths 1 and 2
Measures (edge length 1)
Volume${\displaystyle {\frac {5+4{\sqrt {5}}}{3}}\approx 4.64809}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
3–3: ${\displaystyle \arccos \left({\frac {4{\sqrt {5}}-5}{15}}\right)\approx 74.75474^{\circ }}$
4–4: ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$
Height${\displaystyle {\sqrt {\frac {10-2{\sqrt {5}}}{5}}}\approx 1.05146}$[1]
Central density1
Number of external pieces22
Level of complexity8
Related polytopes
ArmyPobcu
RegimentPobcu
DualDeltotrapezohedral icosahedron
Abstract & topological properties
Flag count160
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×A1, order 20
Flag orbits8
ConvexYes
NatureTame

The pentagonal orthobicupola (OBSA: pobcu) is one of the 92 Johnson solids (J30). It consists of 10 triangles, 10 squares, and 2 pentagons. It can be constructed by attaching two pentagonal cupolas at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the cupolas are joined such that the bases are rotated 36º, the result is the pentagonal gyrobicupola.

## Vertex coordinates

A pentagonal orthobicupola of edge length 1 has vertices given by the following coordinates:

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