Gyroelongated pentagonal cupola

Gyroelongated pentagonal cupola
Type	Johnson J23 - J24 - J25
Faces	3x5+10 triangles 5 squares 1 pentagon 1 decagon
Edges	55
Vertices	25
Vertex configuration	5(3.4.5.4) 2.5(33.10) 10(34.4)
Symmetry group	C5v
Dual polyhedron	-
Properties	convex
Net

In geometry, the gyroelongated pentagonal cupola is one of the Johnson solids (J₂₄). As the name suggests, it can be constructed by gyroelongating a pentagonal cupola (J₅) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal bicupola (J₄₆) with one pentagonal cupola 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]

Area and Volume

With edge length a, the surface area is

${\displaystyle A={\frac {1}{4}}\left(20+25{\sqrt {3}}+\left(10+{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\approx 25.240003791...a^{2},}$

and the volume is

${\displaystyle V=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\approx 9.073333194...a^{3}.}$

Dual polyhedron

The dual of the gyroelongated pentagonal cupola has 25 faces: 10 kites, 5 rhombi, and 10 pentagons.

Dual gyroelongated pentagonal cupola	Net of dual

External links

• Weisstein, Eric W., "Gyroelongated pentagonal cupola" ("Johnson solid") at MathWorld .

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 .