Great Icosahedron Calculator

Calculations at a great icosahedron. This is the fourth of four Kepler-Poinsot polyhedra or regular star polyhedra, which are regular, non-convex polyhedra. The great icosahedron is made from a small stellated dodecahedron with edge length a, where on each side an irregular tetrahedron (s, t, u, u, a, s) is removed, so that equilateral triangles as faces appear.
Enter one value and choose the number of decimal places. Then click Calculate.

	Edge length (a):			20 faces, 180 sides, 30 edges, 180 ridges, 12 peaks Faces: 20 equilateral triangles Sides: 60 isosceles triangles with legs u and base a, 120 triangles with the sides s, t and u
	1. ridge length (s):
	2. ridge length (t):
	3. ridge length (u):
	Circumsphere radius (rc):
	Surface area (A):
	Volume (V):
	Surface-to-volume ratio (A/V):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Length and radius 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 icosahedron is the most complex of the four Kepler-Poinsot polyhedra, small stellated dodecahedron, great stellated dodecahedron, great dodecahedron, and great icosahedron. It is also the one with the greatest number of faces of those. Its dual polyhedron is the great stellated dodecahedron. Like the great dodecahedron, the great icosahedron is a rarely encountered form. While it is sometimes used for decorative purposes, there appear to be no natural occurrences of comparable forms. Its mathematical significance lies particularly in the very fact that such a complicated, regular, non-convex form exists.

The symmetry properties of the great icosahedron correspond to those of the icosahedron. The icosahedron is also the convex hull of the great icosahedron. The edge length of the enclosing icosahedron is φ+1 (equivalent to φ²) times the edge length of the great icosahedron, where φ represents the golden ratio. Thus, the edge length of the great icosahedron is 1/(φ+1), or 1/φ² times that of the enclosing icosahedron, with φ+1=(1+√5)/2+1≈2,61803398875.

Calculation of volume and other dimensions: great icosahedron

Last updated on 06/15/2026. Author: Jürgen Kummer

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