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Platonic Solids Essay I think that there are exactly five regular polyhedra, and I intend to prove why there are exactly five polyhedra. Ok, firstly, we need to identify what the five polyhedra are. They are the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron. All of these are regular polyhedra have something in common. For each shape, each of its faces are the same regular polygon, and the same number of faces …

…shape after a hexagon has angle measurements greater than 120, there cannot be any more regular polyhedra. This is why that there are only five different regular polyhedra. There can never be more polyhedra, or else the faces would be different shapes and/or the number of faces meeting at a vertex would be different. And because this is the rule that all regular polyhedra must have, those other "non-regular" polyhedra can never become regular polyhedra.