Elongated pentagonal gyrocupolarotunda

Elongated pentagonal gyrocupolarotunda
Type	Johnson J40 – J41 – J42
Faces	3×5 triangles 3×5 squares 2+5 pentagons
Edges	70
Vertices	35
Vertex configuration	10(3.43) 10(3.42.5) 5(3.4.5.4) 2.5(3.5.3.5)
Symmetry group	C5v
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids (J₄₁). As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda (J₃₃) by inserting a decagonal prism between its halves. Rotating either the pentagonal cupola (J₅) or the pentagonal rotunda (J₆) through 36 degrees before inserting the prism yields an elongated pentagonal orthocupolarotunda (J₄₀).

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 an elongated pentagonal gyrocupolarotunda

Formulae

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

${\displaystyle V={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 16.936...a^{3}}$

${\displaystyle A={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 33.5385...a^{2}}$

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, "Elongated pentagonal gyrocupolarotunda" from Wolfram Alpha. Retrieved July 25, 2010.

External links

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