Are there any more Regular Solids to be made? Sadly no: and this becomes clear as you build them. Round any vertex the total sum of the angles can’t possibly sum to more than 360 degrees (a complete rotation round a single point); and the combinations we’ve seen here cover all the possible options. The table below shows the numbers and the possible outcomes as you try the many different combinations:
| Flat shape | Angle | Number round each vertex | Total degrees | Shape |
| Triangle | 60 | 3 | 180 | Tetrahedron |
| Triangle | 60 | 4 | 240 | Octahedron |
| Triangle | 60 | 5 | 300 | Icosahedron |
| Triangle | 60 | 6 | 360 | no good: it just comes out flat |
| Square | 90 | 3 | 270 | Cube |
| Square | 90 | 4 | 360 | no good: it just comes out flat |
| Pentagon | 108 | 3 | 324 | Dodecahedron |
| Pentagon | 108 | 4 | 432 | no good: it doesn’t work at all |
| Hexagon | 120 | 3 | 360 | no good: it just comes out flat |
And so it falls out – there are only 5 simple polyhedra. But there are lots more to come if we relax the rules a bit and move on to The 13 Semi-Regular (Archimedean) Solids.
The 5 Regular (Platonic) Solids
The 13 Semi-Regular (Archimedean) Solids