The platonic polydedra:f=2+evF分V;C分E;P分F Tetrahedron Cube Octahedron Dodecahedron Icosahedron .Triangles. The interior angle of an equilateral triangle is 60 degrees thus on a regular polyhedron, only 3, 4, or 5 triangles can meet a vertex. If there were more than 6 their angles would add up to at least 360 degrees which they cant. Consider the possibilities .3 triangles meet at each vertex. This gives rise to a Tetrahedron .4 triangles meet at each vertex. This gives rise to an Octahedron .5 triangles meet at each vertex. This gives rise to an Icosahedron .Squares. Since the interior angle of a square is 90 degrees, at most three squares can meet at a vertex. this is indeed possible and it gives rise to a hexahedron or cube Pentagons. As in the case of cubes, the only possibility is that three pentagons meet at a vertex. This gives rise to a Dodecahedron Hexagons or regular polygons with more than six sides cannot form the faces of a regular polyhedron since their interior angles are at least 120 degreesTetrahedron Cube Octahedron Dodecahedron Icosahedron •Triangles. The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet a vertex. If there were more than 6 their angles would add up to at least 360 degrees which they can't. Consider the possibilities: •3 triangles meet at each vertex. This gives rise to a Tetrahedron. •4 triangles meet at each vertex. This gives rise to an Octahedron. •5 triangles meet at each vertex. This gives rise to an Icosahedron •Squares.Since the interior angle of a square is 90 degrees, at most three squares can meet at a vertex. This is indeed possible and it gives rise to a hexahedron or cube. •Pentagons.As in the case of cubes, the only possibility is that three pentagons meet at a vertex. This gives rise to a Dodecahedron. •Hexagons or regular polygons with more than six sides cannot form the faces of a regular polyhedron since their interior angles are at least 120 degrees. The Platonic Polydedra: f=2+e-v