Pictures of Platonic Solids

Why only five Platonic Solids

The Greeks recognized that there are only five platonic solids. But why is this so?
The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees.

Tetrahedron:
Three triangels at a vertex: 3*60 = 180 degrees

Octahedron:
Four triangles at a vertex: 4*60 = 240 degrees

Icosahedron:
Five triangles at a vertex: 5*60 = 300 degrees

Cube:
Three squares at a vertex: 3*90 = 270 degrees

Dodecahedron:
Three pentagons at a vertex: 3*108 = 324 degrees

Note:
  Six triangles: 6*60 = 360 degrees
  Four squares: 4*90 = 360 degrees
  Four pentagons: 4*108 = 432 degrees
  Three hexagons: 3*120 = 360 degrees
So there are only five Platonic Solids!

*) Regular means that the sides of the polygon are all the same length.
Congruent means that the polygons are all the same size and shape.