Platonic Solids

Five solids, no more, no less — and Euler's polyhedral formula proves it.

The idea

Definition (Platonic solid).

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons, with the same number of faces meeting at every vertex.

Both conditions matter: the first makes the faces alike, and the second makes the vertices alike, so no vertex of the solid can be distinguished from any other.

Theorem.

Exactly five Platonic solids exist: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

Their names count their faces in Greek and end in -hedron, meaning "face": the tetrahedron has $4$ triangular faces, the cube (or hexahedron) $6$ square faces, the octahedron $8$ triangular faces, the dodecahedron $12$ pentagonal faces, and the icosahedron $20$ triangular faces.

The proof that there are only five rests on a counting law that holds for every convex polyhedron.

Theorem (Euler's polyhedral formula).

If a convex polyhedron has $V$ vertices, $E$ edges and $F$ faces, then $V - E + F = 2.$

Ways to work on it

• Walkthrough. Learn the names — tetrahedron, cube, octahedron, dodecahedron, icosahedron.
• Proof. Prove that exactly five regular solids can exist, using Euler's formula.
• Practice. A mystery solid's face data — recover its counts with Euler's formula, then name it.
• Hardest. From the face shape and vertex arrangement alone, derive a solid's vertex, edge, and face counts.

Not sure where to start? Take the ten-question placement test.