Elongated pentagonal orthobicupola

Elongated pentagonal orthobicupola
Type	Johnson J37 – J38 – J39
Faces	10 triangles 2×5+10 squares 2 pentagons
Edges	60
Vertices	30
Vertex configuration	20(3.43) 10(3.4.5.4)
Symmetry group	D5h
Dual polyhedron	-
Properties	convex
Net

In geometry, the elongated pentagonal orthobicupola or cantellated pentagonal prism is one of the Johnson solids (J₃₈). ^[1] As the name suggests, it can be constructed by elongating a pentagonal orthobicupola (J₃₀) by inserting a decagonal prism between its two congruent halves. Rotating one of the cupolae through 36 degrees before inserting the prism yields an elongated pentagonal gyrobicupola (J₃₉).

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. ^[2]

3D model of an elongated pentagonal orthobicupola

Formulae

The following formulae for volume and surface area can be used if all faces are regular, with edge length a: ^[3]

${\displaystyle V={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 12.3423...a^{3}}$

${\displaystyle A=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 27.7711...a^{2}}$

References

1. Weisstein, Eric W. "Elongated Pentagonal Orthobicupola". mathworld.wolfram.com. Retrieved 2023-10-09.
2. 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 .
3. Stephen Wolfram, "Elongated pentagonal orthobicupola" from Wolfram Alpha. Retrieved July 25, 2010.

External links

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