The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of:

The area A and the volume V of the truncated cuboctahedron of edge length a are: The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes.

A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively.

Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components.

Straight lines on the sphere are projected as circular arcs on the plane.

The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left).

The truncated cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

This polyhedron can be considered a member of a sequence of uniform patterns with vertex configuration (4.6.2p) and Coxeter-Dynkin diagram .

For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings.