Regular Dodecahedron Calculator

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 three 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, and that single fact propagates into everything else. Two of the radii come out as tidy expressions in φ:

Midradius: ρ = φ²a/2 exactly, since φ² = (3+√5)/2

Circumradius: R = √3 φ a/2

Divide one by the other and the a cancels along with the halves, leaving R/ρ = √3/φ ≈ 1.0705. The circumradius-to-inradius ratio is less tidy, R/r ≈ 1.2584, with no clean closed form in φ.

The dodecahedron and icosahedron are duals: join the centres of a dodecahedron’s twelve faces and you get an icosahedron’s twelve vertices.

Dihedral angle

The dihedral angle between two adjacent faces is arccos(−1/√5) ≈ 116.565°, which you will also see written as 180° − arctan(2). Note the minus sign in front: arctan(2) on its own is 63.435°, the supplement, and quoting that as the dihedral angle is a common slip.

At 116.565° the faces meet much more gently than a cube’s 90°, which is what makes a dodecahedron sit and roll the way it does.

Which Platonic solid is closest to a sphere?

Not this one, despite what you will read. Compare them with sphericity, which is scale-free and measures how efficiently a shape wraps its volume:

The icosahedron wins, and it is not close. The confusion comes from comparing surface-to-volume ratios at equal edge length, where the dodecahedron looks best only because a dodecahedron of edge 1 is a far bigger object than an icosahedron of edge 1. That comparison measures size, not shape.

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.