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

Figure Triangle

3

(2n - 4) right angles

= (2 × 3 - 4) × 90°

= (6 - 4) × 90°

= 2 × 90°

= 180°

Quadrilateral

Figure Quadrilateral

4

(2n - 4) right angles

= (2 × 4 - 4) × 90°

= (8 - 4) × 90°

= 4 × 90°

= 36

Pentagon

Figure Pentagon

5

(2n - 4) right angles

= (2 × 5 - 4) × 90°

= (10 - 4) × 90°

= 6 × 90°

= 54

Hexagon

Figure Hexagon

6

(2n - 4) right angles

= (2 × 6 - 4) × 90°

= (12 - 4) × 90°

= 8 × 90°

= 72

Heptagon

Figure Heptagon

7

(2n - 4) right angles

= (2 × 7 - 4) × 90°

= (14 - 4) × 90°

= 10 × 90°

= 90

Octagon

Figure Octagon

8

(2n - 4) right angles

= (2 × 8 - 4) × 90°

= (16 - 4) × 90°

= 12 × 90°

= 108


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

Terms Related to Polygons

Interior and Exterior of the Polygon

Convex and Concave Polygons

Regular and Irregular 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 

From Sum of the Interior Angles of a Polygon to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Comparison of Numbers | Compare Numbers Rules | Examples of Comparison

    May 18, 24 02:59 PM

    Rules for Comparison of Numbers
    Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of digits, we start comparing the digits…

    Read More

  2. Numbers | Notation | Numeration | Numeral | Estimation | Examples

    May 12, 24 06:28 PM

    Numbers are used for calculating and counting. These counting numbers 1, 2, 3, 4, 5, .......... are called natural numbers. In order to describe the number of elements in a collection with no objects

    Read More

  3. Face Value and Place Value|Difference Between Place Value & Face Value

    May 12, 24 06:23 PM

    Face Value and Place Value
    What is the difference between face value and place value of digits? Before we proceed to face value and place value let us recall the expanded form of a number. The face value of a digit is the digit…

    Read More

  4. Patterns in Numbers | Patterns in Maths |Math Patterns|Series Patterns

    May 12, 24 06:09 PM

    Complete the Series Patterns
    We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only r…

    Read More

  5. Worksheet on Bar Graphs | Bar Graphs or Column Graphs | Graphing Bar

    May 12, 24 04:59 PM

    Bar Graph Worksheet
    In math worksheet on bar graphs students can practice the questions on how to make and read bar graphs or column graphs. Test your knowledge by practicing this graphing worksheet where we will

    Read More

Polygons - Worksheets

Worksheet on Polygon and its Classification

Worksheet on Interior Angles of a Polygon

Worksheet on Exterior Angles of a Polygon