Elongated square cupola

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]

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

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

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

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

Dual elongated square cupola	Net of dual

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]

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