# Regular Polygon

## Introduction to Regular Polygon

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 polygon here we will learn about regular polygon. In this article, we will learn about regular and irregular polygons, properties of regular polygons along solved examples of them.

## 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.

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

$\frac{(2n-4)}{n}\times90^{o}$

### 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

### 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

 Number of Sides Name Number of Vertices Interior Angles Exterior Angles 3 Equilateral Triangle 3 3 angles  of 60o 3 angles of 120o 4 Square 4 4  angles of 90o 4  angles of 90o 5 Regular Pentagon 5 5 angles of 108o 5 angles  of 72o 6 Regular Hexagon 6 6 angles of 120o 6 angles of 60o 7 Regular Heptagon 7 7 angles of 128.57o 7 angles of 51.43o 8 Regular Octagon 8 8 angles of 135o 8 angles of 45o 9 Regular Nonagon 9 9 angles of 140o 9 angles of 40o 10 Regular Decagon 10 10 angles of 144o 10 angles of 36o

### 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 =  $\frac{(n-2)\times180^{o}}{n}$

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 = $\frac{n(n-3)}{2}$

### Area of a Regular Polygon

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

$Area=\frac{l^{2}\times n}{4tan(\frac{\pi }{n})}$

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.

### Solved Examples

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

Sol: The given parameters are,

l = 4 cm and n = 5

The formula for finding the area is,

$A=\frac{l^{2}\times n}{4tan(\frac{\pi }{n})}$

$A=\frac{5^{2}\times 5}{4tan(\frac{\pi }{n})}$

$A=\frac{125}{5\times0.726}$

$A=\frac{125}{3.63}$

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?

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

Exterior angle = $\frac{360^{o}}{n}$

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

Put the value of n in the above equation

Exterior angle= $\frac{360^{o}}{6}$

=60o

Hence exterior angle regular hexagon = 60o

1. Is a Rectangle a Regular Polygon?

Ans: A regular polygon has all the sides equal. Polygons are figures which have more than two sides. We name polygons according to the number of sides and angles they have. The most familiar polygons are the triangle, the rectangle, the square and so on. Hence a rectangle is a regular polygon.

2. Define Polygon in Maths?

Ans: In Mathematics, a polygon is a closed two-dimensional shape that has straight line segments. It is not a three-dimensional shape. A polygon does not have any curved surface. A polygon should have at least three sides. Each side of the line segment should intersect with another line segment only at its endpoint. Based on the number of sides of a polygon, we can easily identify the shape of the polygon.

3. How to Find the Area of a Polygon With n-Sides?

Ans: To find the area of a polygon that is not regular or its formula is not defined, we split the figure into triangles, squares, trapezium, etc. In this, the purpose is to visualize the given geometry as a combination of geometries for which we know how to calculate the area. Calculate the area for each of the parts and finally add them up to obtain the area of the polygon.

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