Truncated Cuboctahedron Calculator

Name: Truncated Cuboctahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a regular truncated cuboctahedron or great rhombicuboctahedron, an Archimedean solid with three different faces. Its dual body is the hexakis octahedron.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

A = 12 \cdot a^{2} \cdot (2 + \sqrt{2} + \sqrt{3})

V = 2 \cdot a^{3} \cdot (11 + 7 \sqrt{2})

r_{c} = \frac{a}{2} \cdot \sqrt{13 + 6 \sqrt{2}}

r_{m} = \frac{a}{2} \cdot \sqrt{12 + 6 \sqrt{2}}

\frac{A}{V} = \frac{6 \cdot (2 + \sqrt{2} + \sqrt{3})}{a \cdot (11 + 7 \sqrt{2})}

The truncated cuboctahedron is an 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 truncated cuboctahedron was named by Johannes Kepler because it was supposed by him to be formed from a cuboctahedron whose vertices are truncated so that all edges are of equal length. This isn't entirely accurate, however, as the resulting solid is very similar, but with rectangles instead of squares and semiregular hexagons. Nevertheless, the name stuck.
Another common name is great rhombicuboctahedron. The connection between the great rhombicuboctahedron, the cube, and the octahedron is less clear than that of the small rhombicuboctahedron. However, like the small rhombicuboctahedron and the cuboctahedron, it belongs to the same family of symmetry as the cube and the octahedron, namely the cubic symmetry group. Furthermore, the large rhombicuboctahedron lacks the internal rhombi found within the small rhombicuboctahedron. The historically derived name great rhombicuboctahedron therefore suggests a relationship between these solids rather than being an exact geometric description.
Both common names for this shape, truncated cuboctahedron and great rhombicuboctahedron, are thus not perfect, but when they are used, this is precisely the shape being referred to.

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