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. ^[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] $${\displaystyle C={\frac {1}{2}}a{\sqrt {5+2{\sqrt {2}}}}.}$$

References

1. Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem , Texts and Readings in Mathematics, Hindustan Book Agency, p. 84–89, doi:10.1007/978-93-86279-06-4, ISBN   978-93-86279-06-4 .
2. Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR   0290245 .
3. Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
4. 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, S2CID   122006114, Zbl   0132.14603 .
5. Braileanu, Patricia Isabela; Cananau, Sorin; Dobrescu, Tiberiu Gabriel; Pascu, Nicoleta-Elisabeta (2022), "Finite Element Analysis of Metal Structure Based on Elongated Johnson Cupola", Proceedings in Manufacturing Systems, 17 (3): 89–95, ISSN   2067-9238 .

External links

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