Answers for Activity 9
When completed the tables should look like this:
Regular Polyhedra
Name of Shape
Number of Faces
Number of
Vertices
Number of
Edges
Tetrahedron
4
4
6
Cube
6
8
12
Octahedron
8
6
12
Dodecahedron
12
20
30
Icosahedron
20
12
30
Some Irregular Polyhedra
Name of Shape
Number of Faces
Number of
Vertices
Number of
Edges
Square-based Pyramid
5
5
8
Cuboid
6
8
12
Triangular Prism
5
6
9
Hexagonal Prism
8
12
18
1. The relationship between the number of faces, the number of vertices and the
number of edges for all the polyhedra in both tables is a famous relationship
known as Euler’s Formula or Euler’s Rule. Leonhard Euler discovered that for
polyhedra:
The number of faces + The number of vertices = The number of edges + 2
F+V=E+2
If you try it for other polyhedra other than those listed in the tables above, you
will find that it does apply to all polyhedra that children are likely to meet in Key
Stages 1 and 2.
[Note: For some strange, exotic polyhedra (those with holes, those which
intersect themselves and those which have more than one ‘inside’ region)
Euler’s formula has to be modified slightly but primary school children are
extremely unlikely to encounter these.]
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2. The table which shows the numbers of faces, vertices and edges for regular
polyhedra reveals another relationship, one between pairs of regular polyhedra.
The cube and octahedron produce the same three numerical values for F, V and
E. This is because the cube and octahedron are duals of each other. If you were
to draw lines connecting the midpoints of all the faces of a cube you would
produce an octahedron inside the cube. This would mean that the midpoints of
the 6 faces of the cube would become the vertices of the octahedron. This duality
exists because the number of faces of the cube is the same as the number of
vertices of the octahedron.
Similarly, if you drew lines connecting the mid points of the 8 faces of the
octahedron you would produce a cube inside the octahedron and these
midpoints would be the 8 vertices of the cube. The dodecahedron and
icosahedron are duals of each other (for similar reasons). The tetrahedron is
the dual of itself i.e. connecting midpoints of the faces would produce another
tetrahedron inside the original tetrahedron). These relationships can be
illustrated to children using models made from pipe-cleaners, straws or wire.
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