# Elongated pentagonal rotunda

Elongated pentagonal rotunda
Rank3
TypeCRF
Notation
Bowers style acronymEpro
Coxeter diagramofxx5xoxx&#xt
Elements
Faces
Edges5+5+5+5+5+10+10+10
Vertices5+5+10+10
Vertex figures5 rectangles, edge lengths 1 and (1+5)/2
10 irregular tetragons, edge lengths 1, (1+5)/2, 2, 2
10 isosceles triangles, edge lengths (5+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}}{12}}\approx 14.61197}$
Dihedral angles3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }}$
5–4: ${\displaystyle \arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }}$
4–4: 144°
3–5: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
4–10: 90°
Central density1
Number of external pieces27
Level of complexity22
Related polytopes
ArmyEpro
RegimentEpro
DualDecakis order-10 truncated semibisected pentagonal rhombitrapezohedron
ConjugateElongated pentagrammic rotunda
Abstract & topological properties
Flag count220
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
Flag orbits22
ConvexYes
NatureTame

The elongated pentagonal rotunda (OBSA: epro) is one of the 92 Johnson solids (J21). It consists of 5+5 triangles, 5+5 squares, 1+5 pentagons, and 1 decagon. It can be constructed by attaching a decagonal prism to the decagonal base of the pentagonal rotunda.

If a second rotunda is attached to the other decagonal base of the prism in the same orientation, the result is the elongated pentagonal orthobirotunda. If the second rotunda is rotated 36° instead, the result is the elongated pentagonal gyrobirotunda.

## Vertex coordinates

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

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

## Related polyhedra

Three quasi-convex Stewart toroids are made by tunnelling the elongated pentagonal rotunda. Two of these are distinct elongations of the tunnelled pentagonal rotunda.