Elongated square cupola

Elongated square cupola
Type	Johnson J18 – J19 – J20
Faces	4 triangles 13 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 a polyhedron constructed from an octagonal prism by attaching square cupola onto its base. It is an example of Johnson solid.

Construction

The elongated square cupola is constructed from an octagonal prism by attaching a square cupola onto one of its bases, a process known as the elongation. This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon. A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated square cupola is one of them, enumerated as the nineteenth Johnson solid $J_{19}$.

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.

Properties

The surface area of an elongated square cupola $A$ is the sum of all polygonal faces' area. Its volume $V$ can be ascertained by dissecting it into both square cupola and regular octagon, and then adding their volume. Given the elongated triangular cupola with edge length $a$, its surface area and volume are: $${\begin{aligned}A&=\left(15+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\approx 19.561a^{2},\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\approx 6.771a^{3}.\end{aligned}}$$

The dual polyhedron of an 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.)

External links

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