Elongated pentagonal orthobirotunda

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

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (J₄₂). 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).

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (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]

$V=\frac{1}{6}(45+17\sqrt{5}+15\sqrt{5+2\sqrt{5}})a^3\approx21.5297...a^3$

$A=10+\sqrt{30(10+3\sqrt{5}+\sqrt{75+30\sqrt{5}}})a^2\approx39.306...a^2$

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