Tetrakis Hexahedron Calculator

Name: Tetrakis Hexahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a tetrakis hexahedron, the dual body of the truncated octahedron. A tetrakis hexahedron or kiscube is a cube with matching square pyramids attached to its faces. It has six vertices with four edges and eight vertices with six edges. 12 edges, those of the cube, have length a. 24 edges, the slant edges of the pyramids, have length b.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

b = \frac{3}{4} \cdot a

h = \frac{3}{2} \cdot a

A = 3 \cdot a^{2} \cdot \sqrt{5}

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

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

r_{i} = \frac{3}{10} \cdot a \cdot \sqrt{5}

\frac{A}{V} = \frac{2 \cdot \sqrt{5}}{a}

The tetrakis hexahedron is a Catalan solid. The height h is here the distance between two opposite pyramide peaks. Lengths, height and radiuses 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.

In Greek, tetrakis means four times, and hexahedron means a shape with six faces. In this case, four new triangular faces are superimposed onto each of the six sides of a cube. Thus, this shape possesses 24 faces. These faces are isosceles triangles with a base length of a and a leg length of b. The tetrakishexahedron has a volume one and a half times that of the corresponding cube. Consequently, each of the superimposed pyramids accounts for one-twelfth of the cube's volume. This implies that the height of each pyramid is one-quarter of the cube's edge length.
A tetrakishexahedron is a Kleetope, which is a type of polyhedron formed by superimposing a pyramid onto each of the original faces. Other such shapes include, for example, the triakisicosahedron and the small stellated dodecahedron.

The tetrakishexahedron occasionally occurs in nature as a crystal habit, for instance, in fluorite (CaF₂). This form is also used as a 24-sided die. In the realm of design, this shape is of interest in contexts where a cube-like structure featuring additional outward-pointing vertices is desired.

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