Truncated Cube Calculator

Name: Truncated Cube - Geometry Calculator
Author: Jürgen Kummer

Calculations at a truncated cube, or regular truncated hexahedron. A truncated cube is constructed by cutting off the vertices of a cube in a way, so that every new formed edge has the same length. Hereby, the remaining part of the cube's original edge is longer than either of the two cut-off parts. The dual body of the truncated cube is the triakis octahedron.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

a' = a (1 + \sqrt{2})

A = 2 \cdot a^{2} \cdot (6 + 6 \cdot \sqrt{2} + \sqrt{3})

V = \frac{a^{3}}{3} \cdot (21 + 14 \cdot \sqrt{2})

r_{c} = \frac{a}{2} \cdot \sqrt{7 + 4 \cdot \sqrt{2}}

r_{m} = \frac{a}{2} \cdot (2 + \sqrt{2})

\frac{A}{V} = \frac{6 \cdot (6 + 6 \cdot \sqrt{2} + \sqrt{3})}{a \cdot (21 + 14 \cdot \sqrt{2})}

The truncated hexahedron 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 corners of this shape are eight cube corners, where the edge length a of the truncated hexahedron corresponds to the hypotenuse length b of the equilateral triangle of the cube corner.
If identical truncated hexahedrons are placed next to each other in all directions, gaps in the form of regular octahedrons are created. Thus, space can be tiled completely using truncated hexahedrons and matching octahedrons.
The ratio of the edge lengths of a cube and a truncated hexahedron is approximately 2.414 to 1. This ratio is called the silver ratio, analogous to the golden ratio, but far less well-known.

The truncated hexahedron is occasionally used as a design element in architecture and design. Pyrite, iron sulfide (FeS₂), which crystallizes in a cubic shape when unimpeded, can also crystallize in the form of a truncated hexahedron if the growth conditions change during the process. However, this change usually does not affect all vertices. In nanotechnology, certain crystals are grown in precisely this shape to obtain specific chemical and magnetic properties.

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