Elongated pentagonal orthobirotunda

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (J₄₂). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda (J₃₄) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J₆) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda (J₄₃).

Elongated pentagonal orthobirotunda
Type	Johnson J41 - J42 - J43
Faces	2.10 triangles 2.5 squares 2+10 pentagons
Edges	80
Vertices	40
Vertex configuration	20(3.4^2.5) 2.10(3.5.3.5)
Symmetry group	D5h
Dual polyhedron	-
Properties	convex
Net

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]

${\displaystyle V={\frac {1}{6}}(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}})a^{3}\approx 21.5297...a^{3}}$

${\displaystyle A=10+{\sqrt {30(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}}})a^{2}\approx 39.306...a^{2}}$

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 orthobirotunda" from Wolfram Alpha. Retrieved July 26, 2010.

External links

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