Pentagonal rotunda

Pentagonal rotunda
Type	Johnson J5 – J6 – J7
Faces	10 triangles 1+5 pentagons 1 decagon
Edges	35
Vertices	20
Vertex configuration	2.5(3.5.3.5) 10(3.5.10)
Symmetry group	C5v
Rotation group	C5, [5]+, (55)
Dual polyhedron	-
Properties	convex
Net

In geometry, the pentagonal rotunda is one of the Johnson solids (J₆). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.

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]

Formulae

The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a: ^[2]

${\displaystyle V=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\approx 6.91776...a^{3}}$

${\displaystyle {\begin{aligned}A&=\left({\frac {1}{2}}{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\\&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\approx 22.3472...a^{2}\end{aligned}}}$

${\displaystyle R=\left({\frac {1}{2}}\left(1+{\sqrt {5}}\right)\right)a\approx 1.61803...a}$

${\displaystyle H=\left({\sqrt {1+{\frac {2}{\sqrt {5}}}}}\right)a\approx 1.37638...a}$

Dual polyhedron

The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.

Dual pentagonal rotunda	Net of dual

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. "Pentagonal rotunda". Wolfram Alpha Site. Retrieved July 21, 2010.

External links

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