Platonic solid

In geometry, a convex polyhedron bounded by faces which are all the same-sized regular polygon is known as a Platonic solid. There are only five Platonic solids, shown in the table below:

Proving that there are only 5 Platonic solids is rather easy: From the definition, the faces must be regular polygons which can meet three or more at a point with some excess angle, to create a solid angle. Any regular polygon with 7 or more sides cannot meet three or more to a point without overlapping. The regular hexagon can meet three at a point, but with no excess, thus no solid angle is formed. That leaves the regular pentagon, the square, and the equilateral triangle as the only possible faces for a Platonic solid. The regular pentagon and the square can only meet three at a point and have any excess to allow forming a solid angle. The Platonic solids thus formed are the dodecahedron and the cube. The equilateral triangle can meet three, four, or five at a point and form a solid angle; the figures formed are the tetrahedron, octahedron, and icosahedron, respectively.