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Platonic solids have the following characteristics:
In order to do the following problems, you will need Polydrons or other snap-together regular polygons. If you don’t have access to them, print this Shapes PDF document as a template for cutting shapes out of stiff paper or poster board.
a. | Connect three triangles together around a vertex. Complete the solid so that each vertex is the same. What do you notice? Were you able to build a solid? |
b. | Repeat the process with four triangles around a vertex, then five, then six, and so on. What do you notice? |
a. | Connect three squares together around a vertex. Complete the solid so that each vertex is the same. What do you notice? Were you able to build a solid? |
b. | Repeat the process with four squares around a vertex, then five, and so on. What do you notice? |
a. | Connect three pentagons together around a vertex. Complete the solid so that each vertex is the same. What do you notice? Were you able to build a solid? |
b. | Repeat the process with four pentagons, then five, and so on. What do you notice? |
Connect three hexagons together around a vertex. Complete the solid so that each vertex is the same. What do you notice? Were you able to build a solid?
How many Platonic solids are there? Explain why that’s the case.
When working with three-dimensional figures, the terminology can get confusing. (What would you call a side?) It helps if everyone uses the standard names:
The Platonic solids are named for the number of faces they have.
Counting vertices and edges can be tricky. Think about how to “count without counting.” How many faces are there on the polyhedron? How many vertices on each face? How many faces meet at a vertex on the polyhedron? You can put all of this information together to “count” the number of vertices.
For each of the Platonic solids, count the number of vertices, faces, and edges. This is harder than it sounds! Think about how to “count them without counting them.”
Find a pattern in your table. Express it as a formula relating vertices (v), faces (f), and edges (e).
Does your pattern hold for solids other than the Platonic solids? Build several other solids. Count the vertices, faces, and edges, and find out!
a. | Three triangles around a vertex: You will get a figure with four triangular faces. It is called a tetrahedron. |
b. | Four triangles around a vertex: You will get a figure with eight triangular faces. It is called an octahedron.
Five triangles around a vertex: You will get a figure with 20 triangular faces. It is called an icosahedron. Six triangles around a vertex: It lies flat, and you can’t make a solid. |
Problem A2
a. | Follow the instructions. The construction works, and you can construct a cube. A cube uses a total of six squares in the construction. |
b. | It is impossible to make a solid if we try to connect four or more squares at a common vertex. |
a. | Follow the instructions. The construction works, and you can construct a dodecahedron. A dodecahedron has 12 pentagon faces. |
b. | It is impossible to make a solid if we try to connect four or more regular pentagons at a common vertex. |
It is impossible to create a solid by connecting regular hexagons at a common vertex. If you have ever seen a soccer ball, you may have noticed that it is made up of hexagons and pentagons, but not of hexagons alone!
There are only five Platonic solids. The reason is that the sum of the interior angles of the regular polygons meeting at a vertex must not equal or surpass 360°. Otherwise, the figure will lie flat or even fold in on itself.
We need at least three faces to meet at a vertex, but after hexagons, the regular polygons all have interior angles that are more than 120°. So if three of them met at a vertex, there would be more than 360° there. We have already seen that this same angle restriction leads to just three possibilities for triangles (there must be at least three triangles but fewer than six), squares, and pentagons (in each case, there must be at least three but fewer than four).
This leaves a total of just five possible solids that fit the definition of “regular” or “Platonic” solid.
Solid |
Vertices |
Faces |
Edges |
Tetrahedron | 4 | 4 | 6 |
Octahedron | 6 | 8 | 12 |
Icosahedron | 12 | 20 | 30 |
Cube | 8 | 6 | 12 |
Dodecahedron | 20 | 12 | 30 |
The pattern emerging from each row of the table is that, if f is the number of faces, e is the number of edges, and v is the number of vertices, we have v + f = e + 2. (Note that there are many equivalent ways of writing this relationship!)
It turns out that v + f = e + 2 holds for all convex polyhedra. This is known as Euler’s formula, named for one of the most famous mathematicians of all time; he created the formula and proved that it always held.