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 , 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^(∘), 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 . Each interior angle of an equilateral triangle has a measure of 60^(∘). Then, we can consider the following cases.
| Number of triangles that meet at a vertex
|
x= Sum of the interior angles
|
Is x< 360^(∘)?
|
| 3
|
3 * 60^(∘) = 180^(∘)
|
Yes
|
| 4
|
4 * 60^(∘) = 240^(∘)
|
Yes
|
| 5
|
5 * 60^(∘) = 300^(∘)
|
Yes
|
| 6
|
6 * 60^(∘) = 360^(∘)
|
No
|
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 have a measure of 90^(∘) each. With this information, we can consider the following cases.
| Number of squares that meet at a vertex
|
x= Sum of the interior angles
|
Is x< 360^(∘)?
|
| 3
|
3 * 90^(∘) = 270^(∘)
|
Yes
|
| 4
|
4 * 90^(∘) = 360^(∘)
|
No
|
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^(∘) each. Then, we can consider the following cases.
| Number of pentagons that meet at a vertex
|
x= Sum of the interior angles
|
Is x< 360^(∘)?
|
| 3
|
3 * 108^(∘) = 324^(∘)
|
Yes
|
| 4
|
4 * 108^(∘) = 432^(∘)
|
No
|
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^(∘) each.
| Number of hexagons that meet at a vertex
|
x= Sum of the interior angles
|
Is x< 360^(∘)?
|
| 3
|
3 * 120^(∘) = 360^(∘)
|
No
|
In fact, since the interior angles of any regular polygon with 7 or more sides have a measure greater than 120^(∘), 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.
| At each vertex
|
Sum of angle measures at a vertex
|
Solid
|
| 3 triangles meet
|
180^(∘)
|
Tetrahedron
|
| 4 triangles meet
|
240^(∘)
|
Octahedron
|
| 5 triangles meet
|
300^(∘)
|
Icosahedron
|
| 3 squares meet
|
270^(∘)
|
Cube
|
| 3 pentagons meet
|
324^(∘)
|
Dodecahedron
|
