Platonic Solids - Regular Polyhedra

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.