# Elongated pentagonal orthocupolarotunda

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Elongated pentagonal orthocupolarotunda
Rank3
TypeCRF
Notation
Bowers style acronymEpocuro
Coxeter diagramxoxxx5ofxxo&#xt
Elements
Faces5+5+5 triangles, 5+5+5 squares, 1+1+5 pentagons
Edges5+5+5+5+5+5+10+10+10+10
Vertices5+5+5+10+10
Vertex figures5 isosceles trapezoids, edge lengths 1, 2, (1+5}/2, 2
10 rectangles, edge lengths 1 and (1+5)/2
10 trapezoids, edge lengths 1, 2, 2, 2
10 irregular tetragons, edge lengths 1, (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle 5{\frac {11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}}{12}}\approx 16.93602}$
Dihedral angles3–4 rotundaic join: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
3–4 cupolaic: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
4–5 join: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }}$
4–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
4–4 prismatic: 144°
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
3–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }}$
4–4 join: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }}$
Central density1
Number of external pieces37
Level of complexity28
Related polytopes
ArmyEpocuro
RegimentEpocuro
DualPentadeltodecadeltorthodecatetragopentarhombirhombic triacontapentahedron
ConjugateRlongated retrograde pentagrammic orthocupolarotunda
Abstract & topological properties
Flag count280
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal orthocupolarotunda is one of the 92 Johnson solids (J40). It consists of 5+5+5 triangles, 5+5+5 squares, and 1+1+5 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthocupolarotunda.

## Vertex coordinates

An elongated pentagonal orthocupolarotunda of edge length 1 has vertices given by the following coordinates:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),}$
• ${\displaystyle \left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),}$