In this chapter we will learn about interior angles of polygon along with some important formulas related to the topic.

## What are interior angles of polygons ?

The **angles which lie inside the polygon** are called** interior angles**.

The interior angles are formed at the intersection of two sides of polygon.

### Polygon classification and interior angles

Polygons are broadly classified into two categories;

**(a) Regular Polygons**

These polygons have equal sides and equal interior angles.**(b) Irregular Polygons**

These polygons have different sides and interior angle measurement.

### Interior angle formula for regular polygons

The measure of interior angle for regular polygons is given by following formula;

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

Where;

n = number of sides of polygon

## Sum of all interior angle formula for polygon

The measure of sum of all interior angle is given by following formula;

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

Where;

n = number of sides of polygon

## Calculating interior angles of regular polygons

I hope you understood the above mentioned concept. Let us now calculate the interior angle values of different polygons.

Interior Angles of Triangle

It’s a polygon with 3 sides.

Given below is triangle ABC with interior angles ∠A, ∠B & ∠C

#### Calculating interior angle of Triangle

The measure of interior angle is given by following calculation;

\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, each angle of regular triangle **measures 60 degree**.

#### Sum of all angle of Triangle

The sum of all interior angle of triangle is calculated as;

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

### Interior angle of Quadrilateral

It is a polygon with 4 sides.

Given below is the regular quadrilateral with interior angles ∠A, ∠B, ∠C and ∠D.

Calculating Interior angles of Quadrilateral

\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**.

#### Sum of all Interior angle of Quadrilateral

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 4\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 360\ degree}

The sum of all interior angle of quadrilateral measures **360 degree**.

### Interior angles of Pentagon

It’s a polygon with 5 sides.

Given below is the regular polygon with interior angles marked in red color.

#### Calculating Interior angle of Pentagon

The interior angles of regular pentagon is calculated as;

\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**.

#### Sum of all interior angle of Pentagon

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 5\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 540\ degree}

Hence, sum of all interior angles of pentagon measures** 540 degree**.

### Interior angle of Hexagon

It is a polygon with six sides.

Given below is the image of regular hexagon.

Calculating Interior angle of Hexagon

The interior angle of 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\ =\ \frac{4\ \times 180}{6}}\\\ \\ \mathtt{Interior\ angle\ =\ 120\ degree}

Hence, each interior angle of regular polygon measure 120 degree.

#### Sum of all interior angle of Hexagon

The sum of all interior angles of hexagon is calculated as;

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

### Interior angles of Heptagon

It’s a polygon with 7 sides.

Given below is the image of regular heptagon.

Calculating Interior angle of Heptagon

The interior angles of heptagon is measured as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 7-2) \ \times 180}{7}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{5\ \times 180}{7}}\\\ \\ \mathtt{Interior\ angle\ =\ 128.5\ degree}

Hence, each angle of regular heptagon measures **128.5 degree**.

#### Sum of all interior angles of Heptagon

The sum of all interior angles of heptagon is calculated as;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 7\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 600\ degree}

### Interior angles of Octagon

It’s a polygon with 8 sides.

Given below is the image of regular octagon.

Calculating Interior angle of Heptagon

The interior angle of octagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 8-2) \ \times 180}{8}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{6\ \times 180}{8}}\\\ \\ \mathtt{Interior\ angle\ =\ 135\ degree}

Hence, each angle of regular octagon measures **135 degree**.

#### Sum of all interior angles of Heptagon

The sum of interior angles of octagon is calculated as;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 8\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1080\ degree}

### Interior angles of Nonagon

It’s a polygon with 9 sides.

Given below is the image of regular nonagon.

Calculating Interior angle of Nonagon

The interior angle of Nonagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 9-2) \ \times 180}{9}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{7\ \times 180}{9}}\\\ \\ \mathtt{Interior\ angle\ =\ 140\ degree}

Hence, each interior angle measures** 140 degree**.

#### Sum of all interior angles of Nonagon

Sum of all interior angles of Nonagon is calculated as;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 9\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1260\ degree}

### Interior angles of Decagon

It’s a polygon with 10 sides.

Given below is the image of regular decagon with angled marked in red color.

#### Calculating interior angles of Decagon

The interior angle of regular decagon is calculated as;

\mathtt{Interior\ angle\ =\ \frac{( n-2) \ \times 180}{n}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{( 10-2) \ \times 180}{10}}\\\ \\ \mathtt{Interior\ angle\ =\ \frac{8\ \times 180}{10}}\\\ \\ \mathtt{Interior\ angle\ =\ 144\ degree}

Hence. each angle measures **144 degree**.

#### Sum of angles of Decagon

Sum of all interior angles is calculated as;

\mathtt{Sum\ =\ ( n\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ ( 10\ -\ 2) \ \times \ 180}\\\ \\ \mathtt{Sum\ =\ 1440\ degree}