[Maths Class Notes] on Euler’s Formula Pdf for Exam

The fundamental relationship between the trigonometric functions and the complex exponential function has been established by Euler’s formula. Euler’s formula or Euler’s equation is a fundamental equation in mathematics and engineering and can be applied in various ways.

There are two Euler’s formulas in which one is for complex analysis and the other for polyhedra.

Euler’s Formula Equation

Euler’s formula or Euler’s identity states that for any real number x, in complex analysis is given by:

eix = cos x + i sin x,  where 

x = real number

 

e = base of natural logarithm

 

sin x & cos x = trigonometric functions

 

i = imaginary unit.

What is a Polyhedron?

A 3-dimensional solid that is created by joining polygons together is called a Polyhedron, and they are distinguished by the number of faces they have.

 

The polyhedron has the following parts:

 

The face is considered the flat surface that makes up a polyhedron. These faces are regular polygons.

 

Edge is referred to as the regions where the two flat surfaces intersect in order to form a line section.

 

Vertex is the point at which the polyhedron’s edges converge. The Vertex is the corner of a polyhedron and the vertex plurals are referred to as vertices.

Euler’s Characteristics

If all laws are correctly followed, in that case, all the polyhedrons can work with this formula. This formula will work for the majority of the polyhedral. There are several shapes that produce a different response to the FE number. At certain times, the answer to the number of FEs is called the Euler’s Characteristics X. Certain times, it is written as FE = X.  As a negative value for “Double Torus” surface, some shapes have an Euler characteristic. For more complex figures, we can get very complex values.  Euler’s characteristic has a different value for the different shapes and for Polyhedrons.

 

Leonhard Euler was an engineer who made significant and important discoveries in many branches of mathematics, while also making pioneering contributions to many other branches.

 

Leonhard Euler gave a topological invariance which gives the relationship between faces, vertice and edges of a polyhedron. Only for polyhedrons with certain rules, Euler’s Formula works. 

 

The rule is:

• There should be no gaps in the structure.

• It must also not intersect itself.

• It can also not be composed of two parts stuck together, like two cubes stuck by one vertex together.

What is a Polyhedron?

A polyhedron is a 3-dimensional solid that is created by joining polygons together.

 

Two Greek terms originate from the word ‘polyhedron,’ poly meaning several, and hedron referring to surface.

 

Polyhedrons are distinguished by the number of faces they have.

Parts of Polyhedron:

• Face: The faces are considered the flat surfaces that make up a polyhedron. These faces are polygons that are regular. 

• Edge: The edges are referred to as the regions where the two flat surfaces intersect to form a line section.

• Vertex: It is the point at which the polyhedron’s edges converge. The corner of a polyhedron is often referred to as a vertex. Vertex plurals are referred to as vertices.

 

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Define Euler’s Formula

To define the Euler’s formula, it states that the below formula is followed for polyhedrons:

 

F + V – E = 2

 

Where F is the number of faces, the number of vertices is V, and the number of edges is E.

 

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Euler’s Characteristics

If all of the laws are correctly followed, then all polyhedrons can work with this formula. This formula will therefore work for the majority of the common polyhedral.

 

In fact, there are several shapes that produce a different response to the FE number. Sometimes, the answer to the number of FEs is called the Euler’s Characteristics X. Sometimes this is written as FE = X. As a negative value for “Double Torus” surface, some shapes may also have an Euler characteristic. So, for complex figures, it can start to get very complex values.

 

For different shapes/ polyhedrons, the Euler’s characteristic has a different value. It can be represented as:

 

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Euler’s Formula Examples

Look at a polyhedron, for instance, the cube or the icosahedron above, count the number of vertices it has, and name this number V. The cube has 8 vertices, so V = 8. Next, count and name this number E for the number of edges that the polyhedron has. There are 12 edges in the cube, so E = 12 in the case of the cube. Finally, count and mark it F by the number of ears. In the cube example, F = 6.

 

F + V – E = 2

 

When it comes to cube, we know that, 

 

Number of vertices = 8, 

 

Number of edges = 12 and 

 

Number of faces = 6. 

 

So,

 

F + V – E = 6 + 8 – 12

 

F + V – E = 14 – 12

 

F + V – E = 2

 

This is the value Euler’s formula states the cube should have.

Solved Examples

1. How many regular polyhedra convexes are there?

 

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2. Verify Euler’s solid formula.

Solution: Euler’s equation for solids states that,

 

⇒ Faces + Vertices – Edges = 2

 

Since the given shape is Cuboid. Therefore, the number of faces is 6, vertices are 8 and edges are 12.

 

On applying the formula,

 

⇒ 6 + 8 – 12

 

⇒ 14 – 12

 

⇒ 2

3. There are 8 faces and 12 edges of an octahedron. How many vertices has it got?

Solution: Euler’s formula for solids,

 

⇒ Faces + Vertices – Edges = 2

 

We have the value of faces, vertices and edges. On applying, the values to the formula,

 

⇒ 8+Vertices-12=2

 

⇒ Vertices=2+12-8

 

⇒ Vertices=6

 

4. For a cube, prove Euler’s formula.

Solution: Basically to prove Euler’s formula for any polyhedron, we should know that Euler’s characteristics vary on the basis of its number of faces, vertices and edges. And then apply the values to the formula.

 

Therefore, in the case of a cube, the number of faces is 6, the number of vertices is 8, and the number of edges is 12.

 

Therefore, on applying the values to the Euler’s formula, we get,

 

Faces + Vertices – Edges = 2

 

6 + 8 – 12 = 2

 

14 – 12 = 2

 

2= 2

 

Since L.H.S = R.H.S,

 

Hence Proved!