# Pentagonal orthocupolarotunda

Pentagonal orthocupolarotunda Rank3
TypeCRF
SpaceSpherical
Notation
Bowers style acronymPocuro
Coxeter diagramxoxx5ofxo&#xt
Elements
Faces5+5+5 triangles, 5 squares, 1+1+5 pentagons
Edges5+5+5+5+10+10+10
Vertices5+5+5+10
Vertex figures5 isosceles trapezoids, edge lengths 1, 2, (1+5}/2, 2
10 rectangles, edge lengths 1 and (1+5)/2
10 irregular tetragons, edge lengths 1, 2, 1, (1+5)/2
Measures (edge length 1)
Volume$5\frac{11+5\sqrt5}{12} ≈ 9.24181$ Dihedral angles3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$ 4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ 3–5 rotundaic: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$ 3–4 join: $\arccos\left(-\frac{\sqrt{15}-\sqrt3}{6}\right) ≈ 110.95106°$ 3–5 join: $\arccos\left(-\sqrt{\frac{5-2\sqrt5}{15}}\right) ≈ 100.81237°$ Central density1
Related polytopes
ArmyPocuro
RegimentPocuro
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The pentagonal orthocupolarotunda is one of the 92 Johnson solids (J32). It consists of 5+5+5 triangles, 5 squares, and 1+1+5 pentagons. It can be constructed by attaching a pentagonal cupola and a pentagonal rotunda at their decagonal bases, such that the two pentagonal bases are in the same orientation.

If the cupola and rotunda are joined such that the bases are rotated 36º, the result is the pentagonal gyrocupolarotunda.

## Vertex coordinates

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

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

A decagonal prism can be inserted between the two halves of the pentagonal orthocupolarotunda to produce the elongated pentagonal orthocupolarotunda..

The pentagonal orthocupolarotunda also has a connection with the regular icosahedron, being a partial Stott expansion of a pentagonal-symmetric faceting of the icosahedron.