Euler's Formula

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What is Euler’s Formula?

A polyhedron is a 3-dimensional shape that is formed by polygons. Every polygon in a polyhedron is said to be face. The faces meet in a line segment that is called an edge. The point of intersection of edges is called a vertex.We can easily count the number of faces, vertices and edges of a polyhedron. If the number of bases is upto 10. But if the number of sides of base is greater, then counting faces, vertices and edges becomes difficult. To overcome this difficulty, the great Swiss mathematician Leonard Euler discovered a very important relationship among the number of faces(F), vertices(V) and Edges(E) of a polyhedron, called Euler’s formula.

According to Euler’s rule

F + V - E = 2


Euler’s Theorem

  • The Euler theorem is known to be one of the most important mathematical theorems named after Leonhard Euler.

  • Euler’s theorem states a relation between the number of faces, vertices, and edges of any polyhedron. 

  • The Euler’s formula can be written as F + V = E + 2, where F is the equal to the number of faces, V is equal to the number of vertices, and E is equal to the number of edges.

  • The Euler’s formula states that for a polyhedron the number of faces plus the number of vertices minus the number of vertices is equal to 2.


Euler's Formula Proof

Let us Verify Euler’s formula proof for a right triangular prism.

(Image to be added soon)


Solution:

Number of vertices (V) = 6

Number of faces (F) = 5

and number of edges (E) = 9

Euler’s formula:

V + F – E = 2

⇒ 6 + 5 – 9 = 2

⇒ 2 = 2

Hence, the Euler’s formula proof is verified.


Why Always 2? 

When we visualize the euler's formula the question arises why we always get the result is 2.Imagine taking the cube and adding an edge from corner to corner of one face. 

We get one more edge and also extra face.

Now we get face (F) =6 + 1 =  7

Edges(E) = 12 + 1 = 13

vertices(V) = 8

F + V - E = 7 + 8 − 13 = 2

Hence we get the result as 2 only.

Let's take another example: try to include another vertex in a cube and we get an extra edge.

Faces(F) = 6

Edges(E) =12 + 1 = 13

Vertices(V) = 8 + 1 = 9

F + V - E = 6 + 9 − 13 = 2.

No matter what we do, we always end up with result 2.


Euler’s Formula for Other Shapes

Euler’s Formula works for a polyhedron with certain rules that the shape should not have any holes, and also it must not intersect itself. Also, it also cannot be made up of two pieces stuck together, like two cubes stuck together by one vertex.

If all of these rules are properly followed, then this formula will work for all polyhedra.  Thus this formula will work for most of the common polyhedral.

There are in fact many shapes which produce a different answer to the sum F+E-V = 2.  The answer to the sum F+E-V=2 is sometimes called the Euler Characteristic X.

This is often written as F+E-V=X. Some shapes have a Euler Characteristic as negative value as for “Double Torus” surface.  So, it can start to get quite complicated values for complex figures.

This formula can be used in Graph theory. Such as:

  1. To prove a given graph as a planar graph, this formula is applicable.

  2. This formula is very useful to prove the connectivity of a graph.

  3. To find out the minimum colors required to color a given map, with the distinct color of adjoining regions, it is used.

Now let us solve euler's formula examples.


Solved Examples

Euler’s formula examples:

Example 1: A tetrahedron has 4 faces and 4 vertices. How many edges does it have?

Solution: we have,

Faces(F) = 4

Vertices(V) = 4

Edges(E) = ?

A tetrahedron is a Platonic solid, so we can use Euler's formula 

F + V - E = 2 

So substituting the values

 4 + 4 - E = 2

⇒ 8 - E = 2

⇒ E = 8 - 2 = 6

Hence a tetrahedron has 6 edges. 


Example 2: A polyhedron has 6 vertices and 8 faces. How many edges does it have?

Solution:

We can use formula of euler

⇒ F + V - E = 2

Faces(F) = 8,

 Vertices(V) = 6

,Edges(E) = ?

Substitute V = 6 and F = 8

⇒ 8 + 6 - E = 2

⇒ 14 - E = 2

⇒ E = 14 - 2 = 12

Therefore a polyhedron has 12 edges, which is an octahedron:


Quiz Time

  1. A solid has 13 vertices and 24 edges. Using the formula of euler find, how many faces does it have?

  2. An icosahedron has 30 edges and 12 vertices. How many faces does it have?

FAQ (Frequently Asked Questions)

1. What are Faces, Edges and Vertices in a Polyhedron?

  • Faces: The flat 2 - dimensional figures of the solid figures are called the faces of the solid figures. For example, it may be square or rectangular or any polygon.

  • Edges: A line segment between two faces, where the two faces meet are called the edges of the solid shapes.

  • Vertices: A intersecting point of the edges of the solid figures are called vertices. Generally, three faces meet at a single vertex. The plural form of the vertex is vertices.

The below figure represents faces, edges, and vertices of a pyramid, an example of a solid shape.

(Image to be added soon)

2. What is the Relation Between Vertices, Faces and Edges

The relation between vertices, faces and edges can be easily determined with the help of Euler’s Formula. The Euler’s formula holds good for closed solids which have flat faces and straight edges such as the cuboids. It cannot be used for cylinders or cones because they have curved edges.

Euler’s formula is given by

F + V – E = 2

where F, V, and E represents the number of faces, vertices, and edges of the polyhedra respectively.