We will learn how to find the sum of the interior angles of a polygon having n sides.
We know that if a polygon has ‘n’ sides, then it is divided into (n – 2) triangles.
We also know that, the sum of the angles of a triangle = 180°.
Therefore, the sum of the angles of (n – 2) triangles = 180 × (n – 2)
= 2 right angles × (n – 2)
= 2(n – 2) right angles
= (2n – 4) right angles
Therefore, the sum of interior angles of a polygon having n sides is (2n – 4) right angles.
Thus, each interior angle of the polygon = (2n – 4)/n right angles.
Now we will learn how to
find the find the sum of interior angles of different polygons using the
formula.
Name 
Figure 
Number of Sides 
Sum of interior angles (2n  4) right angles 
Triangle 
3 
(2n  4) right angles = (2 × 3  4) × 90° = (6  4) × 90° = 2 × 90° = 180°  
Quadrilateral 
4 
(2n  4) right angles = (2 × 4  4) × 90° = (8  4) × 90° = 4 × 90° = 360°  
Pentagon 
5 
(2n  4) right angles = (2 × 5  4) × 90° = (10  4) × 90° = 6 × 90° = 540°  
Hexagon 
6 
(2n  4) right angles = (2 × 6  4) × 90° = (12  4) × 90° = 8 × 90° = 720°  
Heptagon 
7 
(2n  4) right angles = (2 × 7  4) × 90° = (14  4) × 90° = 10 × 90° = 900°  
Octagon 
8 
(2n  4) right angles = (2 × 8  4) × 90° = (16  4) × 90° = 12 × 90° = 1080° 
Solved examples on sum of the interior angles of a polygon:
1. Find the sum of the measure of interior angle of a polygon having 19 sides.
Solution:
We know that the sum of the interior angles of a polygon is (2n  4) right angles
Here, the number of sides = 19
Therefore, sum of the interior angles = (2 × 19 – 4) × 90°
= (38 – 4) 90°
= 34 × 90°
= 3060°
2. Each interior angle of a regular polygon is 135 degree then find the number of sides.
Solution:
Let the number of sides of a regular polygon = n
Then the measure of each of its interior angle = [(2n – 4) × 90°]/n
Given measure of each angle = 135°
Therefore, [(2n – 4) × 90]/n = 135
⇒ (2n – 4) × 90 = 135n
⇒ 180n – 360 = 135n
⇒ 180n  135n = 360
⇒ 45n = 360
⇒ n = 360/45
⇒ n = 8
Therefore the number of sides of the regular polygon is 8.
● Polygons
Polygon and its Classification
Interior and Exterior of the Polygon
Number of Triangles Contained in a Polygon
Angle Sum Property of a Polygon
Problems on Angle Sum Property of a Polygon
Sum of the Interior Angles of a Polygon
Sum of the Exterior Angles of a Polygon
7th Grade Math Problems
8th Grade Math Practice
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