# Elongated pentagonal gyrocupolarotunda

Elongated pentagonal gyrocupolarotunda
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEpgycuro
Coxeter diagramxoxxo5ofxxx&#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${\displaystyle 5\frac{11+5\sqrt5+6\sqrt{5+2\sqrt5}}{12} ≈ 16.93602}$
Dihedral angles3–4 rotundaic join: ${\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768°}$
3–4 cupolaic: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°}$
4–5 join: ${\displaystyle \arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°}$
4–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
4–4 prismatic: 144°
3–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$
3–4 join: ${\displaystyle \arccos\left(-\sqrt{\frac{10-2\sqrt5}{15}}\right) ≈ 127.37737°}$
4–4 join: ${\displaystyle \arccos\left(-\sqrt{\frac{5-\sqrt5}{10}}\right) ≈ 121.71747°}$
Central density1
Related polytopes
ArmyEpgycuro
RegimentEpgycuro
ConjugateRlongated retrograde pentagrammic fgyrocupolarotunda
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal gyrocupolarotunda is one of the 92 Johnson solids (J41). 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 gyrocupolarotunda.

## Vertex coordinates

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

• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{2},\,0,\,±\frac12\right),}$
• ${\displaystyle \left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+2\sqrt5}{5}}}{2}\right),}$
• ${\displaystyle \left(0,\,-\sqrt{\frac{5+2\sqrt5}{5}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{25+11\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac{3+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,\frac{1+2\sqrt{\frac{5+\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac12,\,\sqrt{\frac{5+2\sqrt5}{20}},\,-\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,-\sqrt{\frac{5+\sqrt5}{40}},\,-\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$
• ${\displaystyle \left(0,\,-\sqrt{\frac{5+\sqrt5}{10}},\,-\frac{1+2\sqrt{\frac{5-\sqrt5}{10}}}{2}\right),}$