[Maths Class Notes] on Regular Polygon Pdf for Exam

In Geometry, a Polygon is a closed two-dimensional figure, which is made up of straight lines. A Polygon is a two-dimensional geometric figure that has a finite number of sides. Generally, from the name of the Polygon, we can easily identify the number of sides of the shape of a polygon. For example, a triangle is a Polygon having three sides. There are different types of Polygons here we will learn about Regular Polygons. In this article, we will learn about Regular and irregular Polygons, properties of Regular Polygons along with solved examples of them.

In other words, a Polygon is a geometric shape with a finite number of sides in two dimensions. A Polygon’s sides are made up of straight line segments that are joined end to end. As a result, a Polygon’s line segments are referred to as sides or edges. The intersection of two line segments is known as the vertex or corner, and an angle is generated as a result. A triangle with three sides is an example of a Polygon. Although a circle is a planar figure, it is not regarded as a Polygon since it is curved and lacks sides and angles. As a result, we may claim that all Polygons are two-dimensional forms, but not all two-dimensional figures are Polygons.

Different sorts of Polygons may be seen in our daily lives, and we may use them deliberately or unknowingly. In this article, you will learn the concept and definition of a Polygon, as well as the many types of Polygons, as well as real-life examples of Polygon forms, their characteristics, and corresponding formulae.

Definition of Regular Polygon

A Polygon having all the sides equal are all Regular Polygons. Since all the sides of the Polygon are equal therefore all the angles of a Regular Polygon are equal.

The most common examples of Regular Polygons are square, rhombus, equilateral triangle etc.

To form a closed figure, a minimum of three line segments must be connected end to end. As a result, a triangle is a Polygon having at least three sides, often known as a 3-gon. The term n-gon refers to a Polygon with n sides.

The below diagram represents a Regular Polygon.

Regular Polygon Examples

• A square has all its sides equal, and all the angles are equal to 90°.

• An equilateral triangle has all three sides equal and the measure of each angle is equal to 60°.

• A Regular pentagon has five equal sides and all the interior angles of the pentagon are equal to 108⁰.

Properties of Regular Polygons

Following are the properties of Regular Polygons:

• All the sides and interior angles of a Regular Polygon are all equal.

• The bisectors of the interior angles of a Regular Polygon meet at its centre.

• The perpendiculars drawn from the centre of a Regular Polygon to its sides are all equal.

• The lines joining the centre of a Regular Polygon to its vertices are all equal.

• The centre of a Regular Polygon is the centre of both the inscribed and circumscribed circles.

• Straight lines drawn from the centre to the vertices of a Regular Polygon divide it into as many equal isosceles triangles as there are sides in it.

• Formula to calculate the angle of a Regular Polygon of n side is                                      =(2n−4n)×90o

Regular and Irregular Polygons

A polygon whose sides are not of the same length and angles are not of the same measure is called Irregular Polygons. In simple words, polygons that do not fulfil the properties of a Regular Polygon are known as Irregular Polygons.

Different Diagram of an Irregular Polygon

Types Of Irregular Polygon

Concave or Convex Polygon

• There are no angles pointing inwards in a convex Polygon. No internal angle can be more than 180 degrees.

• The polygon is concave if any internal angle is bigger than 180°. (Concave has the word “cave” in it.

Simple or Complex Polygon

A simple Polygon has only one border that does not cross over. When a complicated Polygon interacts with itself, it creates a new Polygon! When it comes to Polygons, many rules don’t apply when the situation is complicated.

Parts of Regular Polygons

Following are the five basic parts of a Regular Polygon:

• Vertices

• Sides

• Interior Angles

• Exterior Angles

• Diagonals

The Diagram Shown Below Represent Basic Parts of a Regular Polygon 

Based on the number of sides there are different types of Regular Polygons few of them are listed below

Regular Polygons List

Regular Polygon Angles

Exterior Angles Of A Regular Polygon

Exterior angles of every simple Polygon add up to 360o, because a trip around the Polygon completes a rotation, or return to your starting place. For example in a hexagon where sides meet, they form vertices, so the hexagon has six vertices.

Interior Angles of a Regular Polygon

Inside the hexagon’s sides, where the interior angles are, is called the hexagon’s interior. Outside its sides is the hexagon’s exterior. This becomes important when we consider complex Polygons, like a star-shape (such as pentagram).

Formulas of Regular Polygon

The formula to find the sum of interior angles of a regular Polygon when the value of n is given

The sum of an interior angle = (n-2) x 180⁰

Where n is the number of sides of the Polygon 

The formula to calculate each interior angle of a regular Polygon 

 Interior angle – (n – 2)×180°/n

The formula to calculate each exterior angle of a regular Polygon 

Since all exterior angles sum up to 360°.

Exterior angles = 360⁰/n

To find the sum of  interior and exterior angle of a regular Polygon 

Since each exterior angle is adjacent to the respective interior angle in a regular Polygon and their sum is 180°.

Sum of interior angles + exterior angle = 180⁰

Formula to Find the Number of Diagonals of a Polygon

For an ‘n’-sided Polygon, the number of diagonals can be calculated using the given formula

Number of diagonals = n(n−3)/2n(n−3)2

Area of a Regular Polygon

To calculate the area of a regular Polygon use the below  formula

Area=l2 × n4tan (πn)

Where

l  is the length of any side

n  is the number of sides

tan  is the tangent function calculated in degrees

Formula to Find the Perimeter of a Regular Polygon

The perimeter of a regular Polygon can be calculated with given below formula

Perimeter = ns

where n is the number of sides of the Polygon, s is the measure of one side of the Polygon.

Fun Facts About Polygons

• A Polygon is a two-dimensional geometric shape with at least three straight sides and angles.

• The sides of a regular Polygon are all the same length, and the angles are all the same. The equilateral triangle and square are examples of regular Polygons.

• A triangle is a three-sided Polygon with 180-degree inner angles.

• A square is a four-sided Polygon with 360-degree internal angles. A square is a quadrilateral as well.

• A pentagon is a five-sided Polygon with 540 degrees of inner angles.

• Regular pentagons have equal-length sides and 108-degree interior angles.

• Okra is a pentagonal-shaped dietary plant.

• Hexagonal cells make up beehive cells.

• Curved heptagons are used on the British 50 and 20 penny coins.

Solved Examples

1. Calculate the Area of 5 Sided Regular Polygon Having a Side Length of 5 cm.

Ans: The given parameters are,

l = 4 cm and n = 5

The formula for finding the area is, Area=l2 × n4tan (πn)

A=l2×n4tan(πn)

A=52×54tan(πn)

A=52×54tan(πn)

A=1255×0.726

A=1255×0.726

A=1253.63

A=1253.63

A = 34.43 cm2

Hence the area of a Polygon is 34.43 cm2

2. Calculate the Value of the Exterior Angle of a Regular Hexagon?

Ans:  Here we will use the formula of an exterior angle

Exterior angle = [frac{360^{0}}{n}]

As we a hexagon has 6 sides, so n = 6

Put the value of n in the above equation

Exterior angle =[frac{360^{0}}{6}]

                          =60o

Hence exterior angle regular hexagon = 60o