Platonic Solids Calculator

The five Platonic solids are the only convex polyhedra where every face is the same regular polygon and every vertex is identical. They were known to the ancient Greeks — Plato linked them to the classical elements, though that part did not hold up.

What does hold up is the mathematics. Each solid has exact closed-form formulas for its surface area, volume, and all three radii: the inradius (radius of the inscribed sphere that touches every face), the midradius (radius of the sphere tangent to every edge), and the circumradius (radius of the circumscribed sphere that passes through every vertex).

The formulas for all five solids, in terms of edge length a:

Tetrahedron (4 faces, 4 vertices, 6 edges): SA = sqrt(3)a^2, V = a^3sqrt(2)/12, ri = asqrt(6)/12, rm = asqrt(2)/4, rc = a*sqrt(6)/4

Cube (6 faces, 8 vertices, 12 edges): SA = 6a^2, V = a^3, ri = a/2, rm = asqrt(2)/2, rc = asqrt(3)/2

Octahedron (8 faces, 6 vertices, 12 edges): SA = 2sqrt(3)a^2, V = a^3sqrt(2)/3, ri = asqrt(6)/6, rm = a/2, rc = a*sqrt(2)/2

Dodecahedron (12 faces, 20 vertices, 30 edges): SA = 3sqrt(25+10sqrt(5))a^2, V = (15+7sqrt(5))/4*a^3

Icosahedron (20 faces, 12 vertices, 30 edges): SA = 5sqrt(3)a^2, V = 5(3+sqrt(5))/12a^3

Euler’s formula V - E + F = 2 holds for all five. It is one of the first results in topology and applies to any convex polyhedron.

The dodecahedron and icosahedron are duals of each other: if you place a vertex at the center of each face of one, you get the other. The cube and octahedron are also duals. The tetrahedron is self-dual.

In chemistry, the icosahedron appears in boron compounds and viral capsid structures. The cube is the basis of simple cubic crystal lattices. The tetrahedron appears in carbon bonding geometry (sp3 hybridization produces tetrahedral bond angles of 109.47 degrees).