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 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]

Formulae

The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:^[2]

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

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

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

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]

References

External links

• Weisstein, Eric W., "Elongated square cupola" ("Johnson solid") at MathWorld.