As such, it is a quasiregular polyhedron, i.e., an Archimedean solid that is not only vertex-transitive but also edge-transitive.

[5] The Cartesian coordinates for the vertices of a cuboctahedron with edge length

can be determined by slicing it off into two regular triangular cupolas, summing up their volume.

The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°.

Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5° approximately 125°.

[9] A cuboctahedron has the Rupert property, meaning there is a polyhedron of the same or larger size that can pass through its hole.

[10] The cuboctahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.

[14] Its center is like the apical vertex of a canonical pyramid: one edge length away from all the other vertices.

(In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids).

This radial equilateral symmetry is a property of only a few uniform polytopes, including the two-dimensional hexagon, the three-dimensional cuboctahedron, and the four-dimensional 24-cell and 8-cell (tesseract).

[15] Radially equilateral polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.

Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra.

Because it is radially equilateral, the cuboctahedron's center is one edge length distant from the 12 vertices.

The cuboctahedron can be represented as a configuration matrix with elements grouped by symmetry transitivity classes.

[19] The 24 edges can be partitioned into 2 sets isomorphic to tetrahedral symmetry.

[citation needed] The cuboctahedron shares its skeleton with the two nonconvex uniform polyhedra, the cubohemioctahedron and octahemioctahedron.

These polyhedrons are constructed from the skeleton of a cuboctahedron in which the four hexagonal planes bisect its diagonal, intersecting its interior.

Adding six squares or eight equilateral triangles results in the cubohemicotahedron or octahemioctahedron, respectively.

[20] The cuboctahedron 2-covers the tetrahemihexahedron, which accordingly has the same abstract vertex figure (two triangles and two squares:

)[21] The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point.

This dissection is expressed in the tetrahedral-octahedral honeycomb where pairs of square pyramids are combined into octahedra.