Interior Angles of a Polygon

Interior angles of a polygon are angles within a polygon made by two sides. The interior angles in a regular polygon are always equal. The sum of the interior angles of a polygon can be calculated by subtracting 2 from the number of sides of the polygon and multiplying by 180°. Sum of Interior Angles = (n − 2) × 180°

In this article, we will learn about the interior angles of a polygon, the sum of interior angles of a polygon, the formula for the interior angles, and others in detail.

What is Angle?

Angle measures when the two rays or lines meet at the same point. In other words, the measure of the common point formed by two lines is called an angle.

There are various types of angles some of these are listed below.

• Right Angle
• Acute Angle
• Obtuse Angle

What are Interior Angles of Polygons?

Angles inside the polygon formed by its two sides are known as the interior angles of a polygon. The interior angles are formed by two sides at a vertex of a polygon. The number of interior angles of a polygon is equal to the number of vertices in a polygon.

Interior Angle Definition

Angle formed by the two sides of a polygon inside it is called as the interior angle of polygons. In other words, the angles formed by two sides of a polygon at its vertex within it is referred to as the interior angles of a polygon.

Sum of Interior Angles Formula

The sum of all interior angles of a polygon formula gives us the sum of all the interior angles of the given polygon. The sum of all interior angles of a polygon is given by:

Sum of Interior Angles of a Polygon = (n – 2)×180°

where,

• n is number of sides in polygon

Sum of Interior Angles Formula

Interior Angle Formulas

There are in general there formulas that are used to find interior angles of any polygon that are:

Formula 1:

Formula for interior angle of regular polygon is, here, n is number of sides of regular polygon.

Interior Angles of a Regular Polygon = [180°(n) – 360°] / n

Formula 2:

Formula to find the interior angle of a polygon if exterior angle is given is:

Interior Angle of a Polygon = 180° – Exterior Angle of a Polygon

Formula 3:

If sum of all the interior angles of a regular polygon is given then its interior angle is calculated as:

Interior Angle = Sum of Interior Angles of a Polygon / n

Interior Angles Theorem

Statement: For a polygon of ‘n’ sides, sum of the interior angles is always equal to (n – 2) × 180°.

To prove: Sum of Interior Angles in any Polygon = (2n – 4) right angles

Proof:

Consider a polygon with n sides in which n triangles are formed. ABCDE is a “n” sided polygon shown in the image added below:

Interior Angles Theorem

Take any point O inside the polygon. Join OA, OB, OC, OD and so on.

We know that,

Sum of interior angles of triangle = 180°

Polygons with n sides have n triangles

So, total sum interior angles of n triangles = n × 180°

Sum of interior angles of polygon + Angles at center O = n × 180°…(i)

Sum of angles at center O = 360°

Putting in equation (i)

Sum of interior angles of the polygon = (n × 180°) – 360°

Sum of Interior Angles of Polygon = (n – 2) × 180° = (2n – 4) × 90° = (2n – 4) Right Angles

Hence Proved.

Sum of Interior Angles of a Polygon

Sum of interior angles of various polygon is added in the table below:

Name of Polygon	Number of Interior Angles	Sum of Interior Angles = (n-2) × 180°
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°
Septagon	7	900°
Octagon	8	1080°
Nonagon	9	1260°
Decagon	10	1440°

Interior Angles in Different Types of Polygons

The interior angles and the sum of interior angles in different types of polygons are discussed below.

Interior Angles of Triangle

Since, the triangle has three sides we put the value of n = 3 in the sum of interior angles formula.

Sum of Interior Angles of a Polygon = (n – 2) × 180°

Sum of interior angles of a triangle = (3 – 2) × 180°

Sum of interior angles of a triangle = 180°

Sum of interior angles of triangle is given by:

Sum of Interior Angles of Triangle = 180°

Interior Angles of Quadrilateral

Since, the quadrilateral has four sides we put the value of n = 4 in the sum of interior angles formula.

Sum of interior angles of a quadrilateral = (4 – 2) × 180°

Sum of interior angles of a quadrilateral = 2 × 180°

Sum of interior angles of a quadrilateral = 360°

Sum of interior angles of quadrilateral is given by:

Sum of Interior Angles of a Quadrilateral = 360°

Interior Angles of Pentagon

Since, the pentagons have five sides we put the value of n = 5 in the sum of interior angles formula.

Sum of interior angles of a polygon = (n – 2) × 180°

Sum of interior angles of a pentagon = (5 – 2) × 180°

Sum of interior angles of a pentagon = 3 × 180°

Sum of interior angles of a pentagon = 540°

Sum of interior angles of pentagon is given by:

Sum of Interior Angles of a Pentagon = 540°

Interior Angles of Hexagons

Since, the hexagons have six sides we put the value of n = 6 in the sum of interior angles formula.

Sum of interior angles of a polygon = (n – 2) × 180°

Sum of interior angles of a hexagon = (6 – 2) × 180°

Sum of interior angles of a hexagon = 4 × 180°

Sum of interior angles of a hexagon = 720°

Sum of interior angles of hexagon is given by:

Sum of Interior Angles of a Hexagon = 720°

Interior Angles in Regular Polygons

Polygons with equal sides are called regular polygons. In regular polygons, all the interior angles are equal. The formula for the interior angles and sum of interior angles in regular polygons is given below.

Sum of Interior Angles in Regular Polygon= (n – 2) × 180°

Formula for Each Interior Angle in Regular Polygon

Finding each interior angle in regular polygon is easy and can be found using formula for interior angles in regular polygon. The formula for the same is:

Interior Angle of Regular Polygon = (Sum of all Interior Angles) / Number of Sides in Regular Polygon

or

Interior Angle of Regular Polygon = [(n – 2) × 180°] / n

where,

• n is Number of Sides in Regular Polygon

Interior Angles in Irregular Polygons

Polygons with unequal sides are called irregular polygons. In irregular polygons, all the interior angles are not equal. The formula for the sum of interior angles in irregular polygons is given below.

Sum of Interior Angles in Irregular Polygon= (n – 2) × 180°

where,

• n is Number of Sides in Regular Polygon

Interior and Exterior Angles of a Polygon

The angle formed inside the polygon by its two sides is called as the interior angle of the polygon. The angle formed between a sides and extension of its adjacent side is called as the exterior angle of the polygon.

Difference between Interior and Exterior Angles

Below table represents the difference between interior and exterior angles.

Interior Angles	Exterior Angles
The angle formed at each vertex by two sides of polygon within it is called as interior angle.	The angle formed by one side and by extending adjacent side of a polygon outside it is called as exterior angle.
Interior angles are inside polygon.	Exterior angles are outside polygon.
Sum of interior angles of polygon = (n – 2) × 180°	Sum of exterior angles of polygon = 360°

Exterior Angles

Angles outside the vertices of the polygon are called exterior angles. These angles are formed by one side of the polygon and extension of the other side. The sum of exterior angles of a polygon is always equal to 360 degrees. Measure of each exterior angle of a polygon is calculated as:

Exterior Angle of a Polygon = 360 ÷ Number of Sides

Exterior Angles

The sum of the exterior angle and the relative interior angle is always supplementary, i.e.

Exterior Angle + Interior Angle = 180°

Note

• If the measure of exterior angle of a polygon is given then, to find the measure of interior angle of the polygon we use the formula.

Interior angle of Polygon = 180° – Exterior angle of polygon

• Also, we can find the sum of interior angles of a polygon by finding the number of triangles formed in polygon. The formula for the sum of interior angles of a polygon when the number of triangles formed in polygon is given.

Sum of interior angles of polygon = Number of triangles formed in polygon × 180°

• If in irregular polygon there are m interior angles and measure of m-1 interior angles is given and we have to find the measure of 1 interior angle that is not known, we use the formula for the sum of interior angles in irregular polygon.

Examples on Interior Angles

Example 1: What is the sum of interior angles of a polygon with 10 sides?

Solution:

Sum of Interior Angles of a Polygon = (n – 2) × 180°

Sum of interior angles of a polygon with 10 sides = (10 – 2) × 180°

Sum of interior angles of a polygon with 10 sides = 8 × 180°

Sum of interior angles of a polygon with 10 sides = 1440°

Example 2: If the exterior angle of a polygon is 105° then, find the measure of interior angle of polygon.

Solution:

Exterior angle of a polygon = 105°

To find the measure of interior angle we use the formula.

Interior Angle of Polygon = 180° – Exterior Angle of polygon

Interior angle of Polygon = 180° – 105°

Interior angle of Polygon = 75°

Example 3: Find the interior angle x of below given polygon.

Examples on Interior Angles

Solution:

As the given figure is a quadrilateral and we know that,

Sum of Interior Angles of a Quadrilateral = 360°

100° + 70° + 160° + x = 360°

x + 330° = 360°

x = 360° – 330°

x = 30°

Measure of interior angle x of given irregular polygon = 30°

Example 4: Find the interior angle of a regular polygon with 6 sides.

Solution:

Sum of Interior Angles of a Polygon = (n – 2) × 180°

Sum of interior angles of a regular polygon with 6 sides = (6 – 2) × 180°

Sum of interior angles of a regular polygon with 6 sides = 4 × 180°

Sum of interior angles of a regular polygon with 6 sides = 720°

Interior Angle of Regular Polygon = (Sum of All Interior Angles) / (Number of Sides in Regular Polygon)

Interior angle of a regular polygon with 6 sides = 720°/6

Interior angle of a regular polygon with 6 sides = 120°

Example 5: Find the sum of interior angles of regular pentagon if the number of triangles formed in the regular pentagon is 3.

Solution:

Number of triangles formed in the regular pentagon = 3

Sum of Interior Angles of a Polygon = Number of Triangles Formed in Polygon × 180°

Sum of interior angles of regular pentagon = 3 × 180°

Sum of interior angles of regular pentagon = 540°

FAQs on Interior Angles

Is sum of interior angles of triangle always 180°?

Yes, sum of interior angle of a triangle is always 180°.

What is a 4 sided polygon called?

4 sided polygon is called quadrilateral.

What is the formula for interior angles of a polygon?

Formula for the interior angles of a polygon is:

Interior Angle of a Polygon = (Sum of Interior Angles of Polygon) / (Number of Sides)

How to find interior angles of a polygon?

To find the interior angle of a polygon we first find the sum of interior angles and then divide it by number of sides of polygon.

What is the sum of interior angles of a polygon?

Sum of the interior angles of a polygon is obtained by the formula:

Sum of Interior Angles of a Polygon = (n – 2) × 180°