Gyroelongated pentagonal cupola

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.

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

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