Use pentagonal faces

The next regular polygon we can use for a face of our polyhedron is the regular pentagon. Joining three pentagons at one vertex doesn't really look like anything familiar, but it turns out that if we keep going, attaching pentagons to free edges as we go around the first three pentagons, we will end up with the dodecahedron, the regular solid with three pentagons meeting at every vertex.

And that's where it ends

You can see that three hexagons lie flat in the plane, so no other regular convex polyhedra can exist. The five Platonic solids are the only polyhedra meeting these criteria.