# Elongated square cupola

Elongated square cupola
Type Johnson
J18 - J19 - J20
Faces 4 triangles
3x4+1 squares
1 octagon
Edges 36
Vertices 20
Vertex configuration 8(42.8)
4+8(3.43)
Symmetry group C4v
Dual polyhedron -
Properties convex
Net

In geometry, the elongated square cupola is one of the Johnson solids (J19). As the name suggests, it can be constructed by elongating a square cupola (J4) by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed.

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, surface area and circumradius can be used if all faces are regular, with edge length a:[2]

${\displaystyle V=(3+{\frac {8{\sqrt {2}}}{3}})a^{3}\approx 6.77124...a^{3}}$

${\displaystyle A=(15+2{\sqrt {2}}+{\sqrt {3}})a^{2}\approx 19.5605...a^{2}}$

${\displaystyle C=({\frac {1}{2}}{\sqrt {5+2{\sqrt {2}}}})a\approx 1.39897...a}$

### Dual polyhedron

The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.

Dual elongated square cupola Net of dual

## Related polyhedra and honeycombs

The elongated square cupola forms space-filling honeycombs with tetrahedra and cubes; with cubes and cuboctahedra; and with tetrahedra, elongated square pyramids, and elongated square bipyramids. (The latter two units can be decomposed into cubes and square pyramids.)[3]