The Compendium Geometry is an eBook providing facts, formulas and explanations about geometry.

Platonic Solids - Regular Polyhedra

Convex regular polyhedra are polyhedra composed of equal regular convex faces. These regular polhedra are called Platonic solids.

Euler's Rule: If V is the number of vertices, F the number of faces, and E the number of edges of a convex regular polyhedron then the following equation is valid:

V + F = E + 2

As shown by Euclid there exist only 5 regular convex polyhedra: tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. Following is a table of the basic properties of Platonic solids:

  Faces Edges of Face Vertices Edges at Vertex Edges
Tetrahedron 4 3 4 3 6
Hexhedron (Cube) 6 4 8 3 12
Octahedron 8 3 6 4 12
Dodecahedron 12 5 20 3 30
Icosahedron 20 3 12 5 30

For regular polyhedra the following statements are valid:

  • The vertices of the polyhedron all lie on a sphere.
  • All the vertices are surrounded by the same number of faces.
  • All the vertex figures are regular polygons.
  • All the dihedral angles are equal.
  • All the solid angles are equivalent.

Last Update: 2010-12-06