The five Platonic solids

Among all polyhedra, the regular ones — with faces that are all equal and regular and vertices that are all equal — are surprisingly limited in number.

Three of the five Platonic solids: tetrahedron, cube and octahedron.

There are exactly five regular polyhedra (Platonic solids):

Name	Faces	Vertices	Edges	Face
Tetrahedron	4	4	6	equilateral triangle
Cube	6	8	12	square
Octahedron	8	6	12	equilateral triangle
Dodecahedron	12	20	30	regular pentagon
Icosahedron	20	12	30	equilateral triangle

Observation — Why there are only five

At each vertex the sum of the angles of the faces meeting there must be $<360°$.

• Equilateral triangle ($60°$): $60°\times 3,4,5$ gives $180°,240°,300°$ → tetrahedron, octahedron, icosahedron.
• Square ($90°$): $90°\times 3=270°$ → cube.
• Regular pentagon ($108°$): $108°\times 3=324°$ → dodecahedron.
• Hexagon ($120°$): $120°\times 3=360°$ — too much, the vertex does not “close”.

Hence the regular polyhedra are exactly $5$.

Links

Topics: Synthetic geometry in space
Concepts: Regular polyhedra · Polyhedron · Platonic solids
Skills: Synthetic geometry · Reasoning by cases
People: Plato