The regular polyhedra

Measures and solids, grade 6 Lesson 7 of 16Learning

The regular polyhedra

A regular polygon

A regular polygon is a polygon whose sides are all equal and whose angles are all equal.

An equilateral triangle is a regular triangle. Each of its angles is $\displaystyle 60^\circ$ .
A square is a regular quadrilateral. Each of its angles is $\displaystyle 90^\circ$ .
In a regular pentagon each angle is $\displaystyle 108^\circ$ , and in a regular hexagon $\displaystyle 120^\circ$ .
A rectangle that is not a square is not regular: its angles are equal, but its sides are not.

A regular polyhedron

A regular polyhedron is a solid whose faces are all flat, and which has three properties:

All its faces are regular polygons, and they are all congruent (exactly the same).
The same number of faces meet at every vertex.
It is convex: it has no dents. The segment between any two points of the solid stays inside the solid.

The cube is a regular polyhedron: its six faces are congruent squares, and three faces meet at every vertex.

The five regular polyhedra

There are exactly five regular polyhedra:

	Name	Faces	Number of faces	Faces at each vertex
1	Tetrahedron	equilateral triangles	4	3
2	Cube	squares	6	3
3	Octahedron	equilateral triangles	8	4
4	Dodecahedron	regular pentagons	12	3
5	Icosahedron	equilateral triangles	20	5

The names come from the number of faces, in Greek: tetra – four, octa – eight, dodeca – twelve, icosa – twenty.
The tetrahedron is a triangular pyramid whose faces are all equilateral triangles.
In three of them the faces are equilateral triangles, in one squares, and in one pentagons.

The angles meeting at a vertex

At every vertex of a solid, the angles of the faces meeting there add up to less than $\displaystyle 360^\circ$ – and so there are only five regular polyhedra.

At every vertex of a solid at least three faces meet. Lay the faces that meet at one vertex flat on the table, side by side, around one point:

If their angles together are less than $\displaystyle 360^\circ$ , a gap is left. Fold the faces until the gap closes, and they make a corner of a solid.
If their angles together are exactly $\displaystyle 360^\circ$ , the faces fill the whole turn and lie flat. They cannot be folded into a corner.
If their angles together are more than $\displaystyle 360^\circ$ , the faces do not fit around the point at all.

This is how 3, 4, 5 and 6 equilateral triangles look around a point:

Triangles ( $\displaystyle 60^\circ$ ): three – $\displaystyle 180^\circ$ , four – $\displaystyle 240^\circ$ , five – $\displaystyle 300^\circ$ . Each of these makes a corner: the tetrahedron, the octahedron and the icosahedron. Six triangles are $\displaystyle 360^\circ$ – flat.
Squares ( $\displaystyle 90^\circ$ ): three – $\displaystyle 270^\circ$ , that is the cube. Four squares are $\displaystyle 360^\circ$ – flat.
Pentagons ( $\displaystyle 108^\circ$ ): three – $\displaystyle 324^\circ$ , that is the dodecahedron. Four pentagons are $\displaystyle 432^\circ$ – too much.
Hexagons ( $\displaystyle 120^\circ$ ): three hexagons are already $\displaystyle 360^\circ$ – flat. Polygons with more sides have even larger angles.

So there are only five possibilities, and five regular polyhedra.

Counting edges and vertices

In a regular polyhedron all the faces are equal, so you can count its edges and vertices without seeing it. For example, in a cube:

Edges: the six faces have $\displaystyle {6\times4}={24}$ sides together. Every edge is made of two sides of faces, so there are $\displaystyle {24:2}={12}$ edges.
Vertices: the six faces have $\displaystyle {6\times4}={24}$ corners together. Three corners meet at every vertex, so there are $\displaystyle {24:3}={8}$ vertices.

Exercise 1

How many edges does each solid have?
Use the table of the five regular polyhedra.

The octahedron

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