The word 'interior' means the internal portion or area located inside. Interior angle is formed when two sides of a polygon share an endpoint. The concept of interior angle can be extended to crossed polygon by using the concept of directed angles.

In mathematics, an interior angle is any of the four angles made by a traversal which lie in between the two intersected lines. The angle is measured in degrees or radians.

## Interior Angles of a Hexagon

Hexagon is a polygon having six edges and six vertices.

To find the sum of the interior angles of a hexagon divide the hexagon into triangles as shown in below figure.

There are four triangles, the sum of the angles in each triangle will be 180$^{\circ}$.
So we have 4 $\times$ 180$^{\circ}$ = 720$^{\circ}$.

Therefore the sum of the interior angles of a hexagon is 720$^{\circ}$.

Regular hexagon

In a regular hexagon all sides will be of same length and interior angles will have same size.

As there are six angles in a regular hexagon

=> $\frac{720^{\circ}}{6}$ = 120$^{\circ}$

Each interior angle of a regular hexagon is 120$^{\circ}$.

## Interior Angles of a Polygon

Polygon is a flat shape consisting of straight lines which are joined to form a closed chain and are two dimensional.
The angles inside the polygons are formed by each pair of adjacent sides. So the interior angle of a polygon will have a constant value as it depends on the number of sides only. From the below image we see that the angle formed inside the polygon(regular pentagon) is 108$^{\circ}$.

So 5 $\times$ 108$^{\circ}$ = 540 $^{\circ}$ (Number of sides of a pentagon = 5)
Therefore, the sum of interior angles of this polygon is 540$^{\circ}$.

## Interior Angles of a Pentagon

In geometry, a pentagon is a five sided polygon. Pentagon can be made up of three triangles and the sum of the three interior angles of a pentagon is 540$^{\circ}$.

To find the interior angles of a pentagon we use the following formula:

Sum of the interior angles of a polygon = 180(n - 2) degrees
where n: number of sides
As a pentagon has 5 sides = 5
Therefore when n = 5, Sum of the interior angles of a pentagon = 180 ( 5 - 2 ) = 540$^{\circ}$

## Interior Angles of a Triangle

In geometry triangle is one of the basic shapes, having three vertices and three edges. A triangle is simply a polygon having three sides. Interior angles in a triangle add up to 180$^{\circ}$.

## Interior Angles of a Quadrilateral

Any four sided shape is a quadrilateral having the sides straight. A quadrilateral has four sides and four vertices. The interior angles of a quadrilateral should always add to 360$^{\circ}$.

## Sum of Interior Angles Formula

In any figure the formula to find the sum of the interior angles is given by 180 (n - 2)
where n = Number of sides.
Given below are the simple steps required to find the sum of the interior angles.

1. In a given figure number of sides should be known.
2. Using the formula plug in the given values.
3. Simplify.

## Examples of Interior Angles

### Solved Examples

Question 1: Find the sum of interior angles in a regular decagon and how many degrees is each interior angle?
Solution:

A decagon has 10 sides

The formula to find the sum of interior angles is Sum =  (n - 2) $\times$ 180$^{\circ}$
Plug in n = 10 in the above formula
Sum  = (10 - 2) $\times$ 180$^{\circ}$
Sum = 8 $\times$ 180$^{\circ}$ = 1440

Degree of each interior angle = $\frac{1440}{10}$ = 144

Therefore the sum of interior angles in a regular decagon is 1440$^{\circ}$ and each interior angle = 144$^{\circ}$.

Question 2: Find the sum of interior angles in a nonagon and how many degrees is each interior angle?
Solution:

A  nonagon has 9 sides

The formula to find the sum of interior angles is Sum =  (n - 2) $\times$ 180$^{\circ}$
Plug in n = 9 in the above formula
Sum  = (9 - 2) $\times$ 180$^{\circ}$
Sum = 7 $\times$ 180$^{\circ}$ = 1260

Degree of each interior angle = $\frac{1260}{9}$ = 140

Therefore the sum of interior angles in a regular decagon is 1260$^{\circ}$ and each interior angle = 140$^{\circ}$.

Question 3: If the sum of interior angles in a polygon is 720$^{\circ}$ then how many sides will a polygon have?
Solution:

Given the degree of polygon = 720

The formula to find the sum of interior angles is   (n - 2) $\times$ 180$^{\circ}$
Plug in the given values to find the value of 'n'.

720 = 180 (n - 2)

Divide the above equation by 180
$\Rightarrow$ 4 =  n - 2
$\Rightarrow$ n = 6

If a polygon is of 720$^{\circ}$ then the polygon will have 6 sides.