# Elongated pentagonal rotunda

Elongated pentagonal rotunda
Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymEpro
Coxeter diagramofxx5xoxx&#xt
Elements
Faces5+5 triangles, 5+5 squares, 1+5 pentagons, 1 decagon
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${\displaystyle \frac{45+17\sqrt5+30\sqrt{5+2\sqrt5}}{12} ≈ 14.61197}$
Dihedral angles3–4: ${\displaystyle \arccos\left(-\sqrt{\frac{10+2\sqrt5}{15}}\right) ≈ 169.18768°}$
5–4: ${\displaystyle \arccos\left(-\frac{2\sqrt5}{5}\right) ≈ 153.43495°}$
4–4: 144°
3–5: ${\displaystyle \arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°}$
4–10: 90°
Central density1
Related polytopes
ArmyEpro
RegimentEpro
DualDecakis order-10 truncated semibisected pentagonal rhombitrapezohedron
ConjugateElongated pentagrammic rotunda
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal rotunda 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 36º instead, the result is the elongated pentagonal gyrobirotunda.

## Vertex coordinates

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

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

## Related polyhedra

The three tunnellings of the elongated pentagonal rotunda. The convex hull is shown as a transparency.

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