Cuboctahedron Calculator

Name: Cuboctahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a cuboctahedron. A cuboctahedron is the intersection of cube and octahedron, if both have the same center point and the same orientation, and the edge length of the octahedron is √2 times the edge length of the cube. The dual body of the cuboctahedron is the rhombic dodecahedron.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

A = 2 \cdot a^{2} \cdot (3 + \sqrt{3})

V = \frac{5}{3} \cdot \sqrt{2} \cdot a^{3}

r_{c} = a

r_{m} = \frac{a}{2} \cdot \sqrt{3}

\frac{A}{V} = \frac{18 + 6 \cdot \sqrt{3}}{5 \cdot \sqrt{2} \cdot a}

The cuboctahedron is after the truncated tetrahedron the second Archimedean solid. Edge length and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

The cuboctahedron, along with the icosidodecahedron, is the only convex quasi-regular polyhedron. It is a type of convex polyhedron that is almost, but not quite, as symmetrical as a regular, or Platonic solid. Like the other Archimedean solids, these two have several different types of regular faces. Unlike those, however, they have only one type of edge. In the cuboctahedron, every edge lies between an equilateral triangle and a square. There is no edge between two squares or between two triangles. Therefore, in terms of their degree of symmetry, the cuboctahedron and icosidodecahedron rank between the other Archimedean solids and the five Platonic solids tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Cuboctahedra occur as a crystalline structure in synthetic diamonds, whereas natural diamonds mostly crystallize octahedrally. Cuboctahedra are occasionally found in art and design.
If several identical cuboctahedra are placed next to each other along their square faces in all six directions, gaps in the form of octahedra are created. Cuboctahedra and octahedra together can therefore tile space without gaps. These corresponding cuboctahedra and octahedra have of course the same edge length.

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