Gyroelongated square cupola

In geometry, the gyroelongated square cupola is one of the Johnson solids (J₂₃). As the name suggests, it can be constructed by gyroelongating a square cupola (J₄) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J₄₅) with one square bicupola removed.

Gyroelongated square cupola
Type	Johnson J22 – J23 – J24
Faces	3×4+8 triangles 1+4 squares 1 octagon
Edges	44
Vertices	20
Vertex configuration	4(3.43) 2.4(33.8) 8(34.4)
Symmetry group	C4v
Dual polyhedron	-
Properties	convex
Net

An unfolded gyroelongated square cupola

An unfolded gyroelongated square cupola, faces colored by symmetry

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

The surface area is,

${\displaystyle A=\left(7+2{\sqrt {2}}+5{\sqrt {3}}\right)a^{2}\approx 18.4886811...a^{2}.}$

The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,

${\displaystyle V=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\approx 6.2107658...a^{3}.}$

Dual polyhedron

The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.

Dual gyroelongated square cupola	Net of dual