Regular Dodecahedron Calculator
Calculate volume, surface area, inradius, and circumradius of a regular dodecahedron from its edge length.
Uses exact golden ratio formulas.
A regular dodecahedron has 12 regular pentagonal faces, 20 vertices, and 30 edges. It is one of the five Platonic solids and the one most closely tied to the golden ratio.
The golden ratio φ = (1+√5)/2 ≈ 1.61803 appears throughout the dodecahedron. Each face is a regular pentagon, and the diagonals of those pentagons are in golden ratio to the edges.
Given edge length a:
Volume: V = (15 + 7√5)/4 × a³ ≈ 7.6631 × a³
Surface area: SA = 3√(25 + 10√5) × a² ≈ 20.6457 × a²
Inradius (sphere tangent to all faces): r = a√(250 + 110√5)/20 ≈ 1.1135 × a
Midradius (sphere tangent to all edges): ρ = a(3 + √5)/4 ≈ 1.3090 × a
Circumradius (sphere through all vertices): R = a√3(1 + √5)/4 ≈ 1.4013 × a
All four radii satisfy R : ρ : r ≈ 1.401 : 1.309 : 1.114, a fixed ratio regardless of edge length.
The golden ratio connection
The ratio of a pentagonal face diagonal to its edge is exactly φ. The ratio of the circumradius to the inradius is φ⁴/(√5) approximately. The dodecahedron and icosahedron are duals of each other — the midpoints of a dodecahedron’s faces form an icosahedron.
Dihedral angle
The dihedral angle between two adjacent faces is arctan(2) ≈ 116.565°. This is larger than a cube (90°) and close to a sphere, which is why dodecahedra are used in some geodesic dome designs and why they approximate a ball shape better than any other Platonic solid.
Where it appears
In chemistry, the dodecahedral cage structure appears in clathrate hydrates (methane ice). In nature, quasicrystals discovered in the 1980s have icosahedral/dodecahedral symmetry. The d12 die in tabletop gaming is a regular dodecahedron.
For a unit edge (a = 1): V ≈ 7.663, SA ≈ 20.646, r ≈ 1.114, R ≈ 1.401.