In the last proposition of the Elements Euclid proved that there are only five regular polyhedra, namely the tetrahedron, octahedron, icosahedron, cube, and dodecahedron. To show there can be no more than five he used the fact that in a polyhedra, the sum of the interior angles of the faces which meet at each vertex must be less than 360.
The Mathematics Enthusiast: Vol. 1
, Article 2.
Available at: http://scholarworks.umt.edu/tme/vol1/iss2/2