# 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.

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.

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