What is Euler's Formula?
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!
FAQs on Euler's Formula
1. Explain the use of Euler’s formula?
According to Euler’s Formula,
There can be an ‘n’ number of shapes with an ‘n’ number of sides. There are exceptions here and there are certain rules that need to be followed. Euler's formula says that no simple polyhedron with exactly seven edges exists. In order to find this out, this formula is needed. It can be seen that there is no simple polyhedron with ten faces and seventeen vertices by using Euler's formula in a similar way. Hence, Euler’s is used to find out whether any particular shape can exist or not, by just one step without all the hassle of actually drawing or practically trying out.
2. Define a Polyhedron?
A figure with multiple plane faces, a Polyhedron can also be defined as a three-dimensional solid shape with a certain number of faces, edges and vertices. A polyhedron has a smooth face, straight edges, and sharp corners or vertices. The plural of a polyhedron is Polyhedra. If the faces and vertex figures of Polyhedron are normal (not necessarily convex) polygons, it is said to be regular. There are a total of nine regular polyhedra using this description, five of them are convex Platonic solids and four of them are the concave Kepler-Poinsot solids.
3. Is it possible for Polyhedron to have 20 Edges and 15 Vertices on 10 Faces?
According to the Euler equation, if the Polyhedron has no holes and it does not intersect itself, and if not made with two parts, it can then follow the Euler’s formula, which states that the sum of faces and vertices with the difference of edges has to be equal to 2. In order to check if a particular Polyhedron can exist or not, we use Euler’s formula. With the given Polyhedron which has 20 edges, 15 vertices and 10 faces, and when we have and apply Euler’s formula, the answer we get is not two. Hence, the given Polyhedra is not possible.
4. Explain the Polyhedron types.
Polyhedrons are classified into two types and based on the edges they have.
i) Regular polyhedron
ii) Irregular polyhedron
In the Regular Polyhedron, all the edges are congruent. A regular Polyhedron is made up of regular polygons, and these are also called platonic solids. Examples are the Triangular shaped pyramid and the cube.
An Irregular Polyhedron is formed by the polygons that have a different shape, and where all the elements are different. Here, all sides of the Irregular Polyhedron are not congruent. Examples of Irregular Polyhedrons are the triangular prism and the Octagonal shaped prism.
5. Where can I find important questions and study material on Euler’s Formula?
The topic ‘Euler’s Formula’ is important, especially for the Mathematics and science students, and it is very important to be able to practice some of the important questions so as to understand all concepts and formulas, and to do well in the examinations.
The online portal, Vedantu.com offers important questions along with answers and other very helpful study material on this topic on Euler’s Formula, which have been formulated in a well structured, well researched, and easy to understand manner. These study materials and solutions are all important and are very easily accessible from Vedantu.com and can be downloaded.