Interior Angles of a Polygon

Sum of Interior Angles of a Polygon:

A polygon is a closed geometric figure which has only two dimensions (length and width). All the vertices, sides and angles of the polygon lie on the same plane. Hence it is a plane geometric figure. At the point where any two adjacent sides of a polygon meet (vertex), the angle of separation is called the interior angle of the polygon. A polygon with three sides has 3 interior angles, a polygon with four sides has 4 interior angles and so on. Sum of all the interior angles of a polygon with ‘p’ sides is given as: 

Sum of interior angles = (p - 2) 180°


Sum of Interior Angles of a Polygon Formula:

The formula for finding the sum of the interior angles of a polygon is devised by the basic ideology that the sum of the interior angles of a triangle is 1800. The sum of the interior angles of a polygon is given by the product of two less than the number of sides of the polygon and the sum of the interior angles of a triangle. If a polygon has ‘p’ sides, then 

Sum of interior angles = (p - 2) 180°

Sum of Interior Angles of a Regular Polygon and Irregular Polygon:

A regular polygon is a polygon whose sides are of equal length. Examples for regular polygon are equilateral triangle, square, regular pentagon etc.  An irregular polygon is a polygon with sides having different lengths. Though the sum of interior angles of a regular polygon and irregular polygon with the same number of sides the same, the measure of each interior angle differs. In case of regular polygons, the measure of each interior angle is congruent to the other. However, in case of irregular polygons, the interior angles do not give the same measure. 

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The measure of each interior angle of a regular polygon is equal to the sum of interior angles of a regular polygon divided by the number of sides.

The sum of interior angles of a regular polygon and irregular polygon examples is given below.

Sum of Interior Angles of a Polygon with Different Number of Sides:

Irregular Polygons

Polygon

No. of Sides 

Sum of Interior Angles

Triangle 

3

1800

Quadrilateral

4

3600

Pentagon

5

5400

Hexagon

6

7200

Heptagon 

7

9000

Octagon

8

10800

Nanogen 

9

12600

Decagon

10

14400



Regular Polygons

Polygon

No. of Sides 

Sum of Interior Angles

Measure of Interior Angle

Triangle 

3

1800

600

Quadrilateral

4

3600

900

Pentagon

5

5400

1080

Hexagon

6

7200

1200

Heptagon 

7

9000

128.570

Octagon

8

10800

1350

Nanogen 

9

12600

1400

Decagon

10

14400

1440


Sum of Interior Angles of a Polygon Formula Example Problems:

1. The sum of the interior angles of a regular polygon is 30600. Find the number of sides in the polygon.

Solution: 

Sum of interior angles of a polygon with ‘p’ sides is given by:

Sum of interior angles = (p - 2) 180°

3060° = (p - 2) 180°

p - 2 = \[\frac{3060°}{180°}\]

p - 2 = 17

p = 17 + 2 

p = 19

The polygon has 19 sides.


2. Find the value of ‘x’ in the figure shown below using the sum of interior angles of a polygon formula.

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Solution: 

The figure shown above has three sides and hence it is a triangle. Sum of interior angles of a three sided polygon can be calculated using the formula as:

Sum of interior angles = (p - 2) 180°

60° + 40° + (x + 83)° = (3 - 2) 180°

183° + x = 180°

x = 180° - 183

x = -3

Fun Facts:

  • Polygons are also classified as convex and concave polygons based on whether the interior angles are pointing inwards or outwards.

  • The name of the polygon generally indicates the number of sides of the polygon. 

Prefix in the Name of the Polygon

Number of Sides

Tri

3

Quad

4

Pent

5

Hex

6

Hept

7

Oct

8

Nan

9

Dec

10

FAQ (Frequently Asked Questions)

1. What are Polygons? How are they Classified?

A polygon is a closed geometric figure with a number of sides, angles and vertices. A polygon is a plane geometric figure. Polygons are broadly classified into types based on the length of their sides. If a polygon has all the sides of equal length then it is called a regular polygon. Examples for regular polygons are equilateral triangles and squares. Irregular polygons are the polygons with different lengths of sides. Based on the number of sides, the polygons are classified into several types. A polygon with three sides is called a triangle, a polygon with 4 sides is a quadrilateral, a polygon with five sides is a pentagon, a polygon with 6 sides is a hexagon and so on.

2. What is the Sum of Interior Angles of a Polygon Formula?

Interior angle of a polygon is that angle formed at the point of contact of any two adjacent sides of a polygon. A polygon will have the number of interior angles equal to the number of sides it has. i.e. If a polygon has 5 sides, it will have 5 interior angles. Sum of all the interior angles of a polygon is equal to the product of a straight angle and two less than the number of sides of the polygon. In a regular polygon, all the interior angles measure the same and hence can be obtained by dividing the sum of the interior angles by the number of sides.

Sum of interior angles = (p - 2) 180°