Calculations at a small stellated dodecahedron. This is the first of four Kepler-Poinsot polyhedra or regular star polyhedra, which are regular, non-convex (concave) polyhedra. The small stellated dodecahedron is made from a dodecahedron with edge length a, whose edges are extended so that five meet in one point. As a result, a fitting right pyramid with a regular pentagon as base 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 great stellated dodecahedron.
Enter one value and choose the number of decimal places. Then click Calculate.
Edge a, ridge s and one face in the form of a pentagram P with the chords of length c.
Formulas:
s = a/2 * ( 1 + √5 ) = a * φ
c = a * ( 2 + √5 ) = a + 2s
rc = a/4 * √50 + 22√5
hp = a/5 * √25 + 10√5
A = 15a² * √5 + 2√5
V = 5/4a³ * ( 7 + 3√5 )
Golden ratio phi:
φ = ( 1 + √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 four Kepler-Poinsot solids are the only regular star polyhedra. The first two, the small stellated dodecahedron and the great stellated dodecahedron, have been known for a long time. The small stellated dodecahedron was discovered by Paolo Uccello in 1430, the great stellated dodecahedronr in 1568 by Wenzel Jamnitzer. Both were then described in detail by Johannes Kepler in 1619. The other two, the great dodecahedron and the great icosahedron, were discovered by Louis Poinsot in 1809. The small stellated dodecahedron is dual to the great dodecahedron, the great stellated dodecahedron is dual to the great icosahedron.
