Elongated square cupola explained

Type:	Johnson
Faces:	4 triangles 13 squares 1 octagon
Edges:	36
Vertices:	20
Dual:	-
Properties:	convex
Net:	Johnson solid 19 net.png

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

J₁₉

Properties

The surface area of an elongated square cupola

is the sum of all polygonal faces' area. Its volume

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

, its surface area and volume are:

\begin A &= \left(15+2\sqrt+\sqrt\right)a^2 \approx 19.561a^2, \\ V &= \left(3+\frac\right)a^3 \approx 6.771a^3.\end

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Elongated square cupola".