In this chapter we will learn about polygons with its types and properties.

## Polygons definition

Polygon is a two dimensional geometrical figure with following points;

⟹ it is **made of straight lines**

⟹ two straight lines join at a point called **vertex**

⟹ the straight lines join to **form closed figure**

The following shape is **not a polygon**.

⟹ figure with curved lines

⟹ shape that is not fully closed

## Classification of Polygons

Polygons are classified into two types;

(a) Regular Polygons

(b) Irregular polygons

**(a) Regular Polygons**

The polygon in which the measure of all sides and angles are same is called Regular Polygons.**(b) Irregular Polygon**

The polygon in which has different side length and angle measure are called irregular polygons.

## Types of Polygons

On the **basis of number of sides**, the polygons can be grouped into different types.

Some of the types of polygons are mentioned below;**(a) Triangle**

It’s a polygon with three sides.

Note that in triangle all the sides are straight lines.

There is no diagonal in triangle.

**(b) Quadrilateral**

It is a polygon with 4 sides.

AB, BC, CD & DA are sides of quadrilateral.

There are two diagonals in quadrilateral.

BD and AC are the diagonals.

**(c) Pentagon**

It’s a polygon with 5 sides.

LK, KJ, JI, ID, DL are the sides of polygons.

There are 5 diagonals in pentagon.

The dotted lines show the diagonals.

**(d) Hexagon**

The polygons with 6 sides is known as hexagon.

There are 9 diagonals in Hexagon.

**(e) Heptagon**

Polygon with 7 sides are known as Heptagon.

There are 14 diagonals in heptagon.

**(f) Octagon**

The polygon with 8 sides is called Octagon.

Octagon has 20 diagonals.

**(g) Nonagon**

The polygon with 9 sides is called Nonagon.

It has 27 diagonals.

**(h) Decagon**

Polygon with 10 sides is called decagon.

It has 35 diagonals.

**Note:**

All the shown images are form of regular polygons as the angle and side measurements are equal. If change the side or angle length you will get irregular polygon.

There are infinite types of polygons in geometry. Just change the number of sides and you will get new polygon.

### Diagonals of Polygons

There is a formula to calculate number of diagonals in given polygon.

**The formula is;**

\mathtt{Number\ of\ diagonal\ =\ \frac{n\ ( n-3)}{2}}

Where;

n = number of sides in polygon.

Let us find number of diagonals in each polygons.

**Diagonal of Triangle**

Number of sides n = 3

\mathtt{Diagonals\ =\ \frac{n\ ( n-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{3\ ( 3-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ 0}

There are **no diagonals in triangle**.

**Diagonals in Quadrilateral**

Number of sides (n) = 4

Using the formula;

\mathtt{Diagonals\ =\ \frac{n\ ( n-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{4\ ( 4-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ 2}

There are **2 diagonals in quadrilateral**.

**Diagonals in Pentagon**

Number of sides ( n ) = 5

Using the diagonals formula;

\mathtt{Diagonals\ =\ \frac{n\ ( n-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{5\ ( 5-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{5\ ( \ 2\ )}{2}}\\\ \\ \mathtt{Diagonal\ =\ 5}

There are **5 diagonals in polygon**.

**Diagonals in Hexagon**

Number of sides (n) = 6

Using the diagonal formula;

\mathtt{Diagonals\ =\ \frac{n\ ( n-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{6\ ( 6-3)}{2}}\\\ \\ \mathtt{Diagonal\ =\ \frac{6\ ( \ 3\ )}{2}}\\\ \\ \mathtt{Diagonal\ =\ 9}

Hence, **there are 9 diagonals in hexagon**.

### Interior angles of Regular Polygon

We know that in regular polygon, all the angles and sides are of equal measure.

**The formula for interior angle of polygon is given as;**

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

where;**n = number of sides in polygon**

Let us calculate values of interior angle for some regular polygons.

Interior Angles of triangle

Number of sides (n) = 3

Using the formula;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 3-2) \ \times 180}{3}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{1\ \times 180}{3}}\\\ \\ \mathtt{Interior\ angle\ =\ 60\ degree}

Hence in regular triangle, each angles **measures 60 degrees**.

#### Interior angles of Quadrilateral

Number of sides = 4

Using the interior angle formula;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 4-2) \ \times 180}{4}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{2\ \times 180}{4}}\\\ \\ \mathtt{Interior\ angle\ =\ 90\ degree}

Hence, each angle of regular quadrilateral measures **90 degree**.

#### Interior angles of regular Pentagon

Number of sides (n) = 5

Applying interior angles formula;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 5-2) \ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{3\ \times 180}{5}}\\\ \\ \mathtt{Interior\ angle\ =\ 108\ degree}

Hence, each angle of pentagon measures** 108 degree**