# Triangular orthobicupola

Triangular orthobicupola
Type Johnson
J26 - J27 - J28
Faces 2+6 triangles
6 squares
Edges 24
Vertices 12
Vertex configuration 6(32.42)
6(3.4.3.4)
Symmetry group D3h
Dual polyhedron Trapezo-rhombic dodecahedron
Properties convex
Net

In geometry, the triangular orthobicupola is one of the Johnson solids (J27). As the name suggests, it can be constructed by attaching two triangular cupolas (J3) along their bases. It has an equal number of squares and triangles at each vertex; however, it is not vertex-transitive. It is also called an anticuboctahedron.

A Johnson solid is one of 92 strictly convex regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.

The triangular orthobicupola is the first in an infinite set of orthobicupolae.

The triangular orthobicupola has a superficial resemblance to the cuboctahedron, which would be known as the triangular gyrobicupola in the nomenclature of Johnson solids — the difference is that the two triangular cupolas which make up the triangular orthobicupola are joined so that pairs of matching sides abut (hence, "ortho"); the cuboctahedron is joined so that triangles abut squares and vice versa. Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron. Hence, another name for the triangular orthobicupola is the anticuboctahedron.

The elongated triangular orthobicupola (J35), which is constructed by elongating this solid, has a (different) special relationship with the rhombicuboctahedron.

The dual of the triangular orthobicupola is the trapezo-rhombic dodecahedron. It has 6 rhombic and 6 trapezoidal faces, and is similar to the rhombic dodecahedron.

## Formulae

The following formulae for volume, surface area, and circumradius can be used if all faces are regular, with edge length a:[1]

$V={\frac {5{\sqrt {2}}}{3}}a^{3}\approx 2.35702...a^{3}$

$A=2(3+{\sqrt {3}})a^{2}\approx 9.4641...a^{2}$

The circumradius of a triangular orthobicupola is the same as the edge length (C=a).

## References

1. ^ Stephen Wolfram, "Triangular orthobicupola" from Wolfram Alpha. Retrieved July 23, 2010.