Elongated pentagonal rotunda

Elongated pentagonal rotunda
Type	Johnson J20 - J21 - J22
Faces	2x5 triangles 2x5 squares 1+5 pentagons 1 decagon
Edges	55
Vertices	30
Vertex configuration	10(42.10) 10(3.42.5) 2.5(3.5.3.5)
Symmetry group	C5v
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J₂₁). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J₆) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J₄₂) with one pentagonal rotunda removed.

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

${\displaystyle V={\frac {1}{12}}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 14.612...a^{3}}$

${\displaystyle A={\frac {1}{2}}\left(20+{\sqrt {5\left(145+58{\sqrt {5}}+2{\sqrt {30\left(65+29{\sqrt {5}}\right)}}\right)}}\right)a^{2}\approx 32.3472...a^{2}}$

Dual polyhedron

The dual of the elongated pentagonal rotunda has 30 faces: 10 isosceles triangles, 10 rhombi, and 10 quadrilaterals.

Dual elongated 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. Stephen Wolfram, "Elongated pentagonal rotunda" from Wolfram Alpha. Retrieved July 22, 2010.

External links

• Weisstein, Eric W., "Elongated pentagonal rotunda" ("Johnson solid") at MathWorld .