Sum of the Interior Angles of a Polygon
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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Worksheet on Polygon and its Classification
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