Academic journal article Australian Mathematics Teacher

Euler's Formula

Academic journal article Australian Mathematics Teacher

Euler's Formula

Article excerpt

A polyhedron is a three-dimensional figure with straight-line edges. Leonhard Euler (1707-1783) discovered a mathematical relation between the number of edges (E), the number of faces (F) and the number of vertices (V) for any solid polyhedron. The table below gives the data for four basic polyhedra: the cube, the tetrahedron, the square pyramid and the triangular prism. This Discovery lesson will be aided if you have these four objects to show your students, particularly so that they understand the difference between a prism and a pyramid. They should also realize that pyramids can also have bases of different shapes just as prisms can; for example, a hexagonal pyramid has a hexagon as its base.

polyhedron          V    F    E

cube                8    6   12
tetrahedron         4    4    6
square pyramid      5    5    8
triangular prism    6    5    9

Ask your students if they can suggest a formula that would connect V, F and E. This formula is

V + F - E = 2

Your students should check that the formula still holds for other polyhedra such as the hexagonal prism or the octahedron. Next they could combine basic polyhedra to obtain more complex solids such as a cube with square pyramids on various faces. If every face of the cube was the base for an attached square pyramid, this three-dimensional star-like object would have V = 14, F = 24, E = 36, which again satisfies Euler's formula.

You can create a number of interesting hands-on problems at this stage. …

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