# Regular Polygon

In this chapter we will learn the concept of regular polygon with properties and examples.

Lets us first review the basics of polygon.

## What is Polygon ?

Polygon is 2 dimensional closed figure made of straight lines.

The polygons are named according to the number of sides and angle in the figure.

For example;

Three side polygon is known as triangle.

Four side polygon is known as quadrilateral and so on.

## What are regular polygons?

The polygon with same side length and angle measurement are known as regular polygons.

Important features of regular polygons are;

(a) Regular polygon sides

In regular polygon all the side lengths are equal.

(b) Regular polygon angles

In regular polygon all the interior angles are equal.

(c) Interior angle formula

For regular polygon, the measure of interior angles can be calculated using the following formula;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}

Where;
n = number of sides of polygon

(d) Sum of interior angles

In regular polygon, the sum of interior angles is given by following formula;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}

Where;
n = number of sides of polygon

(e) Exterior angle formula for regular polygon

In regular polygon, the measure of all exterior angles are equal.

The value of exterior angle can be calculated using following formula;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}

Where;
n = number of sides of polygon

(f) Sum of exterior angles

The sum of all the exterior angle is always equal to 360 degree.

(g) Circumcircle of Regular polygon

The circumcircle is the circle that surround the polygon from the outside.

It touches all the vertices of the given polygon.

Given above is the regular octagon ABCDEFGH.

Note that the circle with center O surrounds the given octagon and touches all its vertices.

(h) Incircle of given polygon

The incircle lies inside the polygon and touches all its sides.

Given above is the regular octagon ABCDEFGH with incircle.

Note that the incircle fits perfectly inside the regular octagon and touches all its sides.

(i) Breaking regular polygons

The regular polygons can be broken into small triangles of equal area.

The basic rule is that a polygon with n sides can be divided into n number of equal triangles.

For Example;
Given above is the regular pentagon ABCDE.

Note that the pentagon with 5 sides in divided into 5 equal triangles.

If you can find the are of triangle then calculating the total are of pentagon will be lot easier.

I hope you understood the above concepts, let us now see some examples of regular polygon.

## Regular Polygon Examples

### Regular Triangle

It’s a regular polygon with three sides.

Given above is a regular triangle ABC.

(a) Regular triangle Sides
All the sides are equal.
AB = BC = CA

(b) All interior angles are equal.
∠A = ∠B = ∠C

(c) Interior angle measurement

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 3-2) \ \times 180}{3}}\\\ \\ \mathtt{Interior\ angle\ =\ 60\ degree}

Hence, each interior angle measure 60 degree.

(d) Sum of interior angle

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 3\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 180\ degree}

Hence, sum of all interior angle is 180 degree.

(e) Exterior angle measurement

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{3}}\\\ \\ \mathtt{Exterior\ angle\ =\ 120\ degree}

Hence, in regular triangle each exterior angle measure 120 degree.

(f) Sum of exterior angle

The sum of exterior angle in any regular polygon is 360 degree.

(g) Circumcenter and incenter

In the above diagram note the circumcenter and incenter of triangle.

The circumcenter passes through all vertices and incenter passes through all sides.

(h) Dividing into parts

The regular triangle can be further divided into three triangles.

### Regular Hexagon

Regular Hexagon is a polygon with 6 sides.

(a) Regular hexagon sides

The regular hexagon ABCDEF has equal sides.
Hence, AB = BC = CD = DE = EF = FA

(b) Regular hexagon Angles
In regular hexagon, all angles are of equal measurement.
∠A = ∠B = ∠C = ∠D = ∠E = ∠F

(c) Interior angle measurement

The interior angle of regular hexagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 6-2) \ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ 120\ degree}

Hence, each angle in a regular hexagon measures 120 degree.

(d) Sum of interior angles

The sum of interior angle of regular polygon is calculated as;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 6\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 720\ degree}

Hence in hexagon, sum of all interior angle measures 720 degree.

(f) Exterior angle measurement

The exterior angle is calculated as;

\mathtt{Exterior\ angle\ =\ \frac{360\ }{n}}\\\ \\ \mathtt{Exterior\ angle\ =\ \frac{360}{6}}\\\ \\ \mathtt{Exterior\ angle\ =\ 60\ degree

Hence in regular hexagon, each exterior angle measures 60 degree.

(g) Sum of exterior angle

The sum of exterior angle in any regular polygon is 360 degree.

(h) Circumcenter and incenter of regular hexagon

(i) Dividing into small triangles

The regular hexagon can be divided into 6 equal triangles.