Exercise 36 Page 622

By taking a close look at the five Platonic solids, we can see that at least $3$ faces/angles meet at each vertex.

Since all the faces of the Platonic solids are regular polygons, we have that all the angles that meet at a vertex have the same measure. Also, notice that their sum must be less than $360^{\circ },$ otherwise, the solid flattens out.

Next, let's study the possible faces for a Platonic solid.

Triangular Faces

A regular triangle is the same as an equilateral triangle. Each interior angle of an equilateral triangle has a measure of $60^{\circ }.$ Then, we can consider the following cases.

From the above, we have that there are only three Platonic solids that have triangular faces – namely, the tetrahedron, the octahedron, and the icosahedron.

Square Faces

The interior angles of a square have a measure of $90^{\circ }$ each. With this information, we can consider the following cases.

Thus, there is only one Platonic solid with square faces – namely, the cube.

Pentagonal Faces

The interior angles of a regular pentagon have a measure of $108^{\circ }$ each. Then, we can consider the following cases.

In consequence, there is only one Platonic solid with pentagonal faces, which is the dodecahedron.

Hexagonal Faces

The interior angles of a regular hexagon have a measure of $120^{\circ }$ each.

In fact, since the interior angles of any regular polygon with $7$ or more sides have a measure greater than $120^{\circ },$ we conclude that there are no more Platonic solids than the ones mentioned before.

Summary

As we have seen, there are only five Platonic solids, and this is one way that Plato might have argued it.