Euler’s Formula: Mathematicians have always worked to understand, chart, and measure the world around us. Euler’s formula is an important geometrical concept that provides a way of measuring. It deals with the shape of Polyhedrons which are solid shapes with flat faces and straight edges. A polyhedron, for example, would consist of a cube, whereas a cylinder would not be a polyhedron with curved edges.

Euler’s formula establishes the fundamental relationship between trigonometric functions and complex exponential functions in complex mathematics. It is named after the Swiss mathematician Leonhard Euler. He asserted that 3D shapes are made up of a combination of certain parts. Most of the solid figures consist of polygonal regions. These regions are- faces, edges, and vertices. In this article, we will learn about the application of Euler’s formula in Geometry; read on to learn more.

What is Euler’s Formula?

Swiss mathematician Leonard Euler gave a formula establishing the relation in the number of vertices, edges and faces of a polyhedron which is known as Euler’s Formula, which is as shown. Below:

\(F + V = E + 2\)

Definition of Euler’s Formula

A formula is establishing the relation in the number of vertices, edges and faces of a polyhedron which is known as Euler’s Formula. 

 If \(V, F\) and \(E\) be the number of vertices, number of faces and number of edges of a polyhedron, then,

\(V+F-E-2\)

or

\(F+V=E+2\)

Faces, Edges and Vertices of Solids

Faces, edges, and vertices are called elements of a \(3\)-dimensional shape.

Faces:  Polygon regions forming a solid are called its faces.

Edges: Line segments in which we face forming a solid meet are called its edges.

Vertices: Points of intersection of three faces of a solid are called its vertices.

In the given figure, faces, edges and vertices of a cube have been shown. \(8\) corners of the cube are its vertices. The line segments forming cube are its \(12\) edges. The six square faces forming cube are its \(6\) faces.

Polyhedrons

A solid formed by polygons (polygonal regions) is called a polyhedron.

Terms Related to Polyhedron

1. Faces: The polygons (polygonal regions) forming a polyhedron are called its faces.

2. Edges: The common line segments of two faces (polygonal regions) are called their edges.

3. Vertex: The points of intersection of two edges of a polyhedron are called its vertices. 

Let us look at the following solids.

Each of these solids is formed by polygonal regions, which are its faces. Each pair of faces meet in a line which is edges. The edges of the polyhedron meet in points that are vertices of the polyhedron. 

Note: In a polyhedron, three or more faces may meet at a point. This point is called a vertex.

Many solids are not polyhedrons.

The solids given below are not polyhedrons.

Types of Polyhedrons

The concept of a convex polyhedron is precisely the same as that of a convex polygon.

Convex polyhedron: Convex polyhedron is a polyhedron. The line segment joining any two points inside the polyhedron or on its surface (faces) lies entirely inside or on the polyhedron.

Example.

These are convex polygons exactly.

Concave Polyhedron: A polyhedron that is not a convex polyhedron is called a concave

polyhedron.

These are concave polyhedrons (these are not convex polyhedrons)

Regular polyhedron: A polyhedron formed by regular polygons and having the same number of faces meeting each vertex is called a regular polyhedron.

The above figure is a regular polyhedron. All faces are regular polygons, and the number of faces meeting at each vertex is the same. 

The above figure is not a regular polyhedron. Here all faces are not regular polygons. Also, here number of faces meeting at each vertex is not the same.

At vertex A, three faces meet, but at vertex B., four faces meet.

Two Important Members of the Family of Polyhedrons

Two important members of the family of polyhedrons are prisms and pyramids.

1. Prism: The polyhedron whose base and top are congruent polygons and its other faces (lateral faces) parallelograms is called a prism.

Example:

These are prisms.

2. Pyramid: A polyhedron whose base is a polygon of any number of sides and whose other faces are triangles having a common vertex is called a pyramid.

These are pyramids.

A prism or a pyramid is named according to its base, which is a polygon. Thus, the base of a hexagonal prism is a hexagon and the bottom of a triangular pyramid.

The cuboid is a rectangular prism. The base of a square pyramid is a square.

Terms Related to Prism

1. Regular Prism: A prism whose base and top are regular polygons (polygons having all sides equal) is called a regular prism.

2. Right Prism: A prism whose lateral faces are perpendicular to the base and top of the prism is called a right prism.

3. Axis of a Prism: The line joining the centres of the base and top of a prism is called the axis of the prism.

4. Length of a prism: The length of a prism is the length of the portion of its axis between its base and top.

5. Triangular prism: A prism whose base and top are triangles is called a triangular prism. 

6. Parallelopiped: A prism whose base and top are parallelograms is called a parallelopiped.

Terms Related to Prism Pyramid

1. Vertex: The common vertex of the triangular faces of a pyramid called the vertex of the pyramid.

2. Axis: The axis of a pyramid is the line segment joining its vertex and the centre of its base.

3. Height: The height of a pyramid is the perpendicular length from the vertex to its base.

4. Lateral Edges: The edges through the vertex of a pyramid are called its lateral edges.

5. Lateral Faces: The side triangular faces of a pyramid are called its lateral faces.

6. Right Pyramid: A pyramid is said to be a right pyramid if the perpendicular from its vertex to its base passes through the centre of the base.

7. Regular Pyramid: A pyramid is said to be a regular pyramid if its base is a regular polygon, i.e., a polygon having all sides equal.

8. Slant height: The line segment joining the vertex and the midpoint of a side of the base of the pyramid is called the slant height of the pyramid.

A pyramid is said to be a triangular pyramid if its base is a triangle.

Verification of Euler’s Formula for Some Polyhedrons

1. Cuboid:

\(F = \) number of faces \(= 6\)

\(E = \) number of edges \(= 12\)

\(V = \) number of vertices \(= 8\)

Clearly, \(F+V=E+2\)

\( \Rightarrow 6 + 8 = 12 + 2\)

\( \Rightarrow 14 = 14\)

2. Cube:

\(F = \) number of faces \(= 6\)

\(E = \) number of edges \(= 12\)

\(V = \) number of vertices \(= 8\)

Clearly, \(F+V=E+2\)

\( \Rightarrow 6 + 8 = 12 + 2\)

\( \Rightarrow 14 = 14\)

Solved Examples: Euler’s Formula

Q.1. For cube shape, prove the Euler’s Formula.
Ans: We know in a cube there are \(6\) faces, \(8\) vertices and \(12\) edges

So, \(F = 6, V= 8, E = 12\)
Thus, \(F+V-E\)
\(= 6+8-12\)
\(=2\)
Hence, it is proved.

Q.2. A polyhedron has \(30\) edges and \(20\) vertices, then find the number of its faces?
Ans: From the given,
\(F=\) number of faces \( = ?\)
\(E=\) number of edges \(= 30\)
\(V=\) number of vertices \(= 20\)
Euler’s Formula is given by,
\(F+V=E+2\)
\(⇒F+20=30+2\)
\(⇒F=32-20\)
\(⇒F=12\)
Hence, a polyhedron has \(12\) faces.

Q.3. A polyhedron has \(4\) faces and \(4\) vertices, then find the total number of edges?
Ans: From the given, 
\(F=\) number of faces \(= 4\)
\(E=\) number of edges =?
\(V=\) number of vertices \(=4\)
Euler’s Formula is given by,
\(F+V=E+2\)
\(⇒4+4=E+2\)
\(⇒E=8-2\)
\(⇒E=6\)
Hence, a polyhedron has \(6\) edges.

Q.4. For the shape of a triangular prism, prove the Euler’s Formula.
Ans: We know in a cube there are \(6\) faces, \(8\) vertices and \(12\) edges

So, \(F = 5, V= 6, E = 9\)
Thus, \(F+V-E\)
\(=5+6-9\)
\(= 2\)
Hence, it is proved.

Q.5. A cuboid has \(6\) faces and \(12\) edges, then find the total number of vertices?
Ans: From the given, 
\(F=\) number of faces \( = 6\)
\(E=\) number of edges \( = 12\)
\(V=\) number of vertices =?
Euler’s Formula is given by,
\(F+V=E+2\)
\(⇒6+V=12+2\)
\(⇒V=14-6\)
\(⇒V=8\)
Hence, a cuboid has \(8\) vertices.

Summary

The relation in the number of vertices, edges and faces of a polyhedron gives Euler’s Formula. By using Euler’s Formula, \(V+F=E+2\) can find the required missing face or edge or vertices. In this article, we learnt about polyhedrons, types of polyhedrons, prisms, Euler’s Formula, and how it is verified.

Frequently Asked Questions (FAQ) – Euler’s Formula

Let’s look at some of the commonly asked questions about Euler’s Formula.

Q.1. What is the Formula for faces, edges and vertices?
Ans: Euler’s Formula for faces, edges and vertices is  \(F+V=E+2\).

Q.2. How do you remember Euler’s Formula?
Ans: The Euler’s formula is \(V + F = E + 2\). But, if you look at the Formula backwards, \(2 + E = F + V\), now the numbers and letters are in alphabetical order. This little trait will help you to remember the formula easily.

Q.3. What is Euler’s Formula for three-dimensional figures?
Ans: In Euler’s Formula, \(V, F\) and \(E\) be the number of vertices, number of faces and number of edges of a polyhedron, then
\(V+F-E-2=0\)
or
\(F+V=E+2\)

Q.4. What are the edges of a solid?
Ans: Line segments in which we face forming a solid meet are called its edges.

Q.5. What are the vertices of a solid?
Ans: Points of intersection of three faces of a solid are called its vertices

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