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Platonic solids: volume, surface area and radii

Calculate the volume, surface area, circumscribed and inscribed radii of a regular tetrahedron, cube, octahedron, dodecahedron or icosahedron from its edge length.

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Volume

61.305

In cubic units of the edge length

Surface area

82.583

Radius of the circumscribed sphere (through the vertices)
2.8025
Radius of the inscribed sphere (touching the faces)
2.2270
Dihedral angle between faces
116.57 °
Faces
12
Edges
30
Vertices
20
Euler's formula: vertices − edges + faces = 2

The five Platonic solids

A regular polyhedron, or Platonic solid, is a solid whose faces are all identical regular polygons, with the same number of faces meeting at every vertex. There are exactly five:

Solid Faces Face shape Edges Vertices
Tetrahedron 4 Triangle 6 4
Cube 6 Square 12 8
Octahedron 8 Triangle 12 6
Dodecahedron 12 Pentagon 30 20
Icosahedron 20 Triangle 30 12

Euclid proved that no others exist: the angles of the faces meeting at a vertex must add up to less than 360°, which only allows these five combinations. Plato associated them with the elements in his dialogue Timaeus, hence the name.

The formulas

For an edge length a, each quantity is a constant times a power of a:

  • Volume: tetrahedron a³ ÷ (6√2), cube a³, octahedron a³√2 ÷ 3, dodecahedron (15 + 7√5) ÷ 4 × a³, icosahedron 5(3 + √5) ÷ 12 × a³.
  • Surface area: tetrahedron √3 a², cube 6a², octahedron 2√3 a², dodecahedron 3√(25 + 10√5) a², icosahedron 5√3 a².
  • Circumradius is the radius of the sphere passing through all the vertices, and inradius the radius of the sphere touching all the faces.
  • The dihedral angle is the angle between two adjacent faces.

Example

A dodecahedron with an edge of 2 units:

  • Volume: 61.305 cubic units
  • Surface area: 82.583 square units
  • Circumradius: 2.8025, inradius: 2.2270
  • Dihedral angle: 116.57°
  • 12 faces, 30 edges, 20 vertices

Euler’s formula

For every one of these solids, vertices − edges + faces = 2. For the dodecahedron, 20 − 30 + 12 = 2. Leonhard Euler stated the relation in 1758, and it holds for any convex polyhedron.

Where you meet them

  • Dice for tabletop games use all five shapes, known as d4, d6, d8, d12 and d20.
  • Many viruses have an icosahedral shell, and some crystals and molecules take the shape of tetrahedra or octahedra, such as methane.
  • Geodesic domes and soccer balls start from an icosahedron.
  • The cube and the octahedron are duals: joining the centers of the faces of one gives the other. The dodecahedron and the icosahedron are duals too, and the tetrahedron is its own dual.

Frequently asked questions

What are the Platonic solids?

The five convex solids whose faces are identical regular polygons with the same number meeting at every vertex: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. Euclid proved in his Elements that no others exist.

Why are there only five?

At least three faces must meet at each vertex, and the angles meeting there must add up to less than 360°. That allows three, four or five triangles, three squares or three pentagons, and nothing else.

What is the volume of a dodecahedron?

V = (15 + 7√5) ÷ 4 × a³, about 7.663 × a³, where a is the edge length. A dodecahedron with 2-unit edges has a volume of about 61.3 cubic units and a surface area of about 82.6 square units.

Where do Platonic solids appear?

In tabletop game dice (d4, d6, d8, d12 and d20), in crystals such as pyrite cubes and octahedral diamonds, in virus shells with icosahedral symmetry, and in molecules such as methane, whose atoms form a tetrahedron.

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Results are estimates provided for information purposes. They do not replace professional advice.

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