Pentagonal orthocupolarotunda

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. ^[1]

3D model of a pentagonal orthocupolarotunda

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a: ^[2]

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

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

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 .