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# Euler's Formula

In online geometry help ,An important formula was framed by the Swiss mathematician Leonhard Euler around 1750 A.D. Euler spent most of his life time in Germany and Russia and ultimately he died in Russia.

Amongst many of his wonderful works in mathematics and physics, the formula he framed in geometry is one of the important and interesting formulas in mathematics help.

This formula relates to the number of vertices, edges and faces of a polyhedron.

Let us take a closer look.

## Euler’s Formula – the Formula

In any polyhedron the numbers of vertices (V), edges (E) and faces (F) are governed bythe following formula.

V – E + F = 2

A number of mathematicians proved this formula in different methods. Let us examine the formula in a few cases.

In a tetrahedron, V = 4, E = 6 and F= 4. V – E + F = 4 – 6 + 4 = -2 + 4 = 2

In a cube, V = 8, E = 12 and F= 6. V – E + F = 8 – 12+ 6 = -4 + 6 = 2

In a octahedron, V = 6, E = 12 and F= 8. V – E + F = 6 – 12 + 8 = -6 + 8 = 2

In any polyhedron, the number of faces is the sum of the faces of all the polygons forming the three dimensional shape. Since each polygon shares a common edge with the adjoining polygon, the total number of edges in the polyhedron is half the number of edges of all the polygons. Thus computing the number of faces and the number of edges we can figure out the number of vertices.

## Euler’s Formula – Practical Applications

In many practical applications the number of vertices is an important consideration, given the type of construction of the polyhedron.

The following free geometry examples will explain the practical applications of Euler’s formula.

Example 1

A antenna is build by a hexagonal pyramid placed on a same size hexagonal prism. The efficient functioning of the antenna depends on the number of vertices. How many vertices will help the functioning of the antenna?

Solution

The polygons used for construction are 6 triangles (the hexagonal pyramid), 6 rectangles (the periphery of of hexagonal prism) and an hexagon (the base).

Therefore, in 6 rectangles, 6 triangles and in1 hexagon

F = 6 + 6 + 1 = 13

E = 24 + 18 + 6 = 48/2 = 24

Using Euler’s formula,

V – E + F = 2

V = E - F + 2 = 24 – 13 + 2 = 11 + 2 = 13

So, the number of vertices in the antenna would be 13.

Example 2

The above shape is called a Bucky ball. It consists of 12 pentagons and 20 hexagons.

This shape is very predominant in the field of Chemistry. Also this shape is followed for making soccer balls as it gives the maximum number of vertices.

How many vertices are there in a Bucky ball?

Solution

The polygons used for construction are 12 pentagons and 20 hexagons.

Therefore, in 12 pentagons and 20 hexagons,

F = 12 + 20 = 32

E = (60 + 120) = 180/2 = 90

Using Euler’s formula,

V – E + F = 2

V = E - F + 2 = 90 – 32 + 2 = 58 + 2 = 60

So, the number of vertices in the Bucky ball would be 60.