What’s the easiest way to explain why only 5 polyhedrons exist?

Question

What’s the easiest way to explain why only 5 polyhedrons exist?

In: Mathematics

Answer 1

The faces must be regular polygons but it’s easy to see that the only possibilities are triangles, squares and pentagons. Hexagons just make a flat surface and beyond that doesn’t work at all.

A polyhedron needs at least three faces meeting at each vertex, and:

* for triangles, 3, 4 and 5 per vertex give the tetrahedron, octahedron and icosahedron; 6 triangles make a flat surface and beyond that can’t work;
* for squares, 3 per vertex give a cube; 4 squares make a flat surface and beyond that can’t work; and
* for pentagons, 3 per vertex given a dodecahedron and beyond that can’t work.