Elongated square cupola

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.

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

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.^[1] This cupola covers the octagonal face so that the resulting polyhedron has four equilateral triangles, thirteen squares, and one regular octagon.^[2] It can also be constructed by removing a square cupola from a rhombicuboctahedron, which would also make it a diminished rhombicuboctahedron. 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 ${\displaystyle J_{19}}$.^[3]

Properties

The surface area of an elongated square cupola ${\displaystyle A}$ is the sum of all polygonal faces' area. Its volume ${\displaystyle V}$ can be ascertained by dissecting it into both a square cupola and a regular octagon, and then adding their volume. Given the elongated triangular cupola with edge length ${\displaystyle a}$, its surface area and volume are:^[4] $${\displaystyle {\begin{aligned}A&=\left(15+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\approx 19.561a^{2},\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\approx 6.771a^{3}.\end{aligned}}}$$ Its circumradius ${\displaystyle C}$ is:^[5] $${\displaystyle C={\frac {1}{2}}a{\sqrt {5+2{\sqrt {2}}}}.}$$