Stellated Octahedron Calculator

Name: Stellated Octahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a stellated octahedron (stella octangula). This is a regular octahedron with a matching tetrahedron attached to each of its faces. It can also be seen as two intersecting tetrahedra, which are point symmetric to each other. The edge length a of the stellated octahedron is similar to the edge length of the two intersecting tetrahedra, the edge length of the spikes (ridges) b is similar to the edge length of the original octahedron.
Enter a or b and choose the number of decimal places. Then click Calculate.

Formulas:

a = 2 \cdot b

r_{c} = \frac{a}{4} \cdot \sqrt{6}

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

V = \frac{a^{3}}{8} \cdot \sqrt{2}

Lengths and radius have the same unit (e.g. meter), the surface 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 stellated octahedron is a deltahedron. That is a polyhedron whose faces consist solely of equilateral triangles. It is non-convex, meaning it contains re-entrant regions. The term concave is sometimes used here as the antonym of convex, though in a strictly geometric sense, this term lacks a precise definition.
In geometry, a star generally refers to an object featuring points or spikes. Such an object may be two-dimensional, like a pentagram, or the hexagram, which is the two-dimensional projection of the stellated octahedron. The nomenclature for these shapes derives from the stars in the night sky. Although celestial stars do not, in reality, possess actual spikes, being instead oblate spheroids, the appearance of such spikes arises from optical effects occurring within the lens of the eye or within optical instruments.

The earliest known drawing of a stella octangula dates back to Leonardo da Vinci and the year 1509. One hundred years later, it was given the name stella octangula by Johannes Kepler, while he was studying the geometric properties of star polyhedra.

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