Great Stellated Dodecahedron Calculator

Name: Great Stellated Dodecahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a great stellated dodecahedron. This is the second of four Kepler-Poinsot polyhedra or regular star polyhedra, which are regular, non-convex (concave) polyhedra. The great stellated dodecahedron is made from an icosahedron with edge length a, whose edges are extended so that three meet in one point. As a result, a fitting right pyramid with an equilateral triangle as base, a regular tetrahedron, is attached to each of its faces. The sides of the pyramid are isosceles triangles, the ratio of ridge s to edge a is that of the golden ratio, like in the pentagram b to c. s, c and A are the same as at the small stellated dodecahedron.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

s = \frac{a}{2} \cdot (1 + \sqrt{5}) = a \cdot φ

c = a \cdot (2 + \sqrt{5}) = a + 2 s

r_{c} = \frac{a}{4} \cdot \sqrt{3} \cdot (3 + \sqrt{5})

h_{p} = \frac{a}{6} \cdot \sqrt{3} \cdot (3 + \sqrt{5})

A = 15 a^{2} \cdot \sqrt{5 + 2 \sqrt{5}}

V = \frac{5}{4} a^{3} \cdot (3 + \sqrt{5})

Golden ratio phi:

φ = \frac{1 + \sqrt{5}}{2} = 1.618033988749895...

Length, radius and height have the same unit (e.g. meter), surface areas have 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 great stellated dodecahedron was first discovered in 1568 by the goldsmith and engraver Wenzel Jamnitzer. Johannes Kepler described it in 1619 together with the small stellated dodecahedron. Louis Poinsot discovered the other two regular star polyhedra, the great dodecahedron and the great icosahedron, in 1809. In 1859, these four bodies were named Kepler-Poinsot polyhedra by the mathematician Arthur Cayley and have had this name ever since.
Great stellated dodecahedron are sometimes found as decorative objects, for example as illuminated Christmas stars.

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