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Pentagonal orthocupolarotunda
Type	Johnson J31 – J32 – J33
Faces	3×5 triangles 5 squares 2+5 pentagons
Edges	50
Vertices	25
Vertex configuration	10(3.4.3.5) 5(3.4.5.4) 2.5(3.5.3.5)
Symmetry group	C5v
Dual polyhedron	-
Properties	convex
Net

In geometry, the pentagonal orthocupolarotunda is one of the Johnson solids (J₃₂). As the name suggests, it can be constructed by joining a pentagonal cupola (J₅) and a pentagonal rotunda (J₆) along their decagonal bases, matching the pentagonal faces. A 36-degree rotation of one of the halves before the joining yields a pentagonal gyrocupolarotunda (J₃₃).

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a:

${\displaystyle V={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\approx 9.24181...a^{3}}$

${\displaystyle A=\left(5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\approx 23.5385...a^{2}}$

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Stephen Wolfram, "Pentagonal orthocupolarotunda" from Wolfram Alpha. Retrieved July 24, 2010.

External links

• Weisstein, Eric W., "Pentagonal orthocupolarotunda" ("Johnson solid") at MathWorld.

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