# Platonic Solids

## Objective

Investigate the five platonic solids to understand why they are unique and why there are only five.

## Base Difficulty

This project is pretty easy. But, it's probably best to do if you have or are taking geometry.

## Concept

Take a look at the five solids pictured below.

These solids share some characteristics that make them unique.

Before we go any further, let's think about some geometry.

A polyhedron is a three dimensional solid with faces that are all polygons. Polygons are closed shapes with straight sides. Look a little closer at the five platonic solids. What observation can you make about the faces of each individual solid?

Each face is made of sides that are of equal length. It follows then that all the interior angles are equal. When polygons have this property they are called regular polygons. When a polyhedron is constructed of identical regular polygons, it is called a regular polyhedron.

The points on each solid where the faces meet are called the vertices. A vertex must meet certain requirements. At least three faces must meet at each vertex and the sum of the interior angles of the regular polygons meeting at each vertex cannot meet or exceed 360°.

Let's recap. Platonic solids are regular polyhedrons. This means that they are solids formed from at least three regular polygons meeting at a vertex. Every face is the same and every side of each face is also the same. Because the faces all have equal sides, they also have equal interior angles.

## Hypothesis

Now that you know a little more about platonic solids, why do you think that there are only five? Perhaps you think there are more than five?

## Materials

- Access to a printer
- Paper
- Scissors
- Glue or tape
- Coloring utensils, if desired

## Procedure

Before going any further, it would probably be a good idea to construct your own models of the five platonic solids. Provided below are the patterns for the Tetrahedron and Icosahedron, create your own patterns for the other shapes. If you would like, first color the patterns. Then, cut along the solid outer lines and fold the inner solid lines. Use the tabs to glue or tape the solid together.

Next, use your models to make observations and fill out this chart. NOTE: To find the measure of the interior angle of each polygon face:

- Let n be the number of sides of the polygon.
- Use 180°(n-2) to give the sum of the interior angles.
- Divide by n to give the measure of each individual angle.

Tetrahedron | Hexahedron | Octahedron | Dodecahedron | Icosahedron | |
---|---|---|---|---|---|

Shape of polygon face | Equilateral Triangle | ||||

Number of interior angles on each polygon face | 3 | ||||

Interior angle of polygon face | 60° | ||||

Number of faces (entire solid) | 4 | ||||

Number of edges (entire solid) | 6 | ||||

Number of vertices (entire solid) | 4 | ||||

Faces meeting at vertex | 3 |

Now that you have examined the properties of the five platonic solids, let's try to find out why there are only five.

- Start by choosing a regular polygon. For example, let's take a look at the equilateral triangle.
- If we know that at least three faces are needed to make a vertex, what would happen if three equilateral triangles met at a vertex? Would it be allowed? Remember that the interior angles of the faces meeting at each vertex cannot be greater than or equal to 360°.
- To check, try this...(use your chart as a reference) 60°(degrees of interior angle) x 3(faces meeting at vertex) = 180° 180° < 360°, therefore this is a possibility.
- Which platonic solid is this?
- Now, think about what would happen if 4 equilateral triangles met at each vertex. Would this be allowed? If it is, which platonic solid would it be?
- How about 5 equilateral triangles, 6 equilateral triangles? What happens?
- Next, try the same thing with other regular polygons. (i.e. squares, pentagons, hexagons..)

## Analysis

Were you able to produce all five of the platonic solids?

## Conclusion

This is your chance to show how much you've learned. Be sure to give an answer as to why there are only five platonic solids.

## Extension

Perhaps this project is not quite challenging enough for you or you would like to investigate further into three dimensional solids. Some suggestions would be to consider solids where every face is not the same. What about polyhedron made up of two different types of faces? Are these solids limited in the same way the platonic solids are? Do you think it would be possible to create a solid of all non regular faces? Research polyhedron and let your own curiosity lead you to developing a unique science fair project.