Elongated square cupola

Elongated square cupola
Type	Johnson J18 - J19 - J20
Faces	4 triangles 1+3.4 squares 1 octagon
Edges	36
Vertices	20
Vertex configuration	8(4^2.8) 4+8(3.4^3)
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]

$V=(3+\frac{8\sqrt{2}}{3})a^3\approx6.77124...a^3$

$A=(15+2\sqrt{2}+\sqrt{3})a^2\approx19.5605...a^2$

$C=(\frac{1}{2}\sqrt{5+2\sqrt{2}})a\approx1.39897...a$

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

Elongated square cupola with Square cupola(Johnson solid No.4) substitute for Rhombicuboctahedron on the space filling combinations.

Johnson solid No.19(Elongated square cupola) fill the space with Tetrahedron and Cube.
Johnson solid No.19 fill the space with Cube and Cuboctahedron.
Johnson solid No.19 fill the space with Tetrahedron and Johnson solid No.8(Elongated square pyramid) and Johnson solid No.15(Elongated square bipyramid).
Johnson solid No.19 fill the space with Tetrahedron and Cube and Johnson solid No.1(Square pyramid).
Johnson solid No.19 fill the space with Tetrahedron and Cube and Johnson solid No.1 and either or combination of [Johnson solid No.8,Johnson solid No.15].^[3]

References

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.
2. Stephen Wolfram, "Elongated square cupola" from Wolfram Alpha. Retrieved July 22, 2010.
3. http://woodenpolyhedra.web.fc2.com/J19.html

External links

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