Dodecahedron Calculator

What is a Regular Dodecahedron?

A regular dodecahedron is one of the five Platonic solids — a convex polyhedron with 12 regular pentagon faces, 30 equal edges, and 20 vertices. Three pentagons meet at every vertex. Because the regular pentagon is itself defined by the golden ratio φ = (1 + √5)/2, every property of the dodecahedron is expressed in terms of √5 and φ.

The dodecahedron is the dual of the icosahedron: connecting the 12 face centers of a dodecahedron produces the 20 vertices of an icosahedron, and vice versa. Both share the same icosahedral symmetry group (I_h). Euler's formula confirms a valid convex polyhedron: V − E + F = 20 − 30 + 12 = 2.

How to Use the Dodecahedron Calculator

Select what you want to calculate from the Solve For dropdown. Enter the edge length of the dodecahedron in any unit. The calculator will compute the volume, surface area, circumradius, inradius, and midradius. When solving for edge from volume, enter the known volume instead. All results update instantly and include step-by-step calculation details.

Key Formulas

The regular dodecahedron formulas all involve the golden ratio φ = (1 + √5)/2:

Volume: V = a³ (15 + 7√5) / 4 ≈ 7.6631 × a³
Surface Area: S = 3a² √(25 + 10√5) ≈ 20.6457 × a²
Circumradius: R = a √3 · φ / 2 ≈ 1.4013 × a
Inradius: r = a √((25 + 11√5)/10) / 2 ≈ 1.1135 × a
Midradius: ρ = a φ² / 2 ≈ 1.3090 × a
Edge from Volume: a = ∛(4V / (15 + 7√5))

The numeric coefficient (15 + 7√5)/4 ≈ 7.6631, so for a = 2, V = 2(15 + 7√5) ≈ 61.305 cubic units.

Properties

Faces: 12 regular pentagons
Edges: 30 (all equal length)
Vertices: 20 (three pentagons meet at each)
Dihedral angle: arccos(−1/√5) ≈ 116.565°
Symmetry: Icosahedral (I_h)
Dual polyhedron: Icosahedron

Applications

Tabletop Gaming: The d12 die used in Dungeons & Dragons is a regular dodecahedron
Chemistry: The dodecahedrane molecule (C₂₀H₂₀) places carbon atoms at the vertices
Crystallography: Pyrite crystals grow as pyritohedral (near-dodecahedral) forms
Art and Architecture: Featured in Plato's Timaeus, Leonardo da Vinci's drawings, and Escher prints

Frequently Asked Questions

How do you calculate the volume of a dodecahedron?

V = a³ (15 + 7√5) / 4, where a is the edge length. The numeric coefficient (15 + 7√5)/4 ≈ 7.6631, so V ≈ 7.6631 × a³. For a = 2, V = 2(15 + 7√5) ≈ 61.305 cubic units.

What is the formula for surface area of a dodecahedron?

S = 3a² √(25 + 10√5). Each of the 12 regular pentagon faces has area (a²/4) √(25 + 10√5), and 12 × (1/4) = 3. The coefficient 3 √(25 + 10√5) ≈ 20.6457, so S ≈ 20.6457 × a².

How many faces, edges, and vertices does a dodecahedron have?

A regular dodecahedron has 12 regular pentagon faces, 30 equal edges, and 20 vertices. Three pentagons meet at every vertex. Euler's formula V − E + F = 20 − 30 + 12 = 2 confirms it is a valid convex polyhedron.

How is the dodecahedron related to the golden ratio?

The golden ratio φ = (1 + √5)/2 ≈ 1.618 appears in every key formula. The midradius is exactly ρ = a φ²/2, the circumradius is R = a √3 · φ/2, and the volume and surface area both involve √5 (since √5 = 2φ − 1).

What is the difference between a dodecahedron and an icosahedron?

They are dual Platonic solids. A dodecahedron has 12 pentagonal faces, 30 edges, and 20 vertices; an icosahedron has 20 triangular faces, 30 edges, and 12 vertices — the face and vertex counts are swapped. Connecting the face centers of one produces the vertices of the other.

What is the circumradius of a regular dodecahedron?

R = a · √3 · φ / 2 ≈ 1.4013 · a, where φ = (1 + √5)/2 is the golden ratio. This is the radius of the sphere that passes through all 20 vertices. For a = 2, R = √3 · φ ≈ 2.803.

Report