[Maths Class Notes] on Interior Angles of a Polygon Pdf for Exam

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 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) 1800

 

Sum of Interior Angles of a Regular Polygon and Irregular Polygon

A regular polygon is a polygon whose sides are of equal length. Examples of a 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 is the same, the measure of each interior angle differs. In the case of regular polygons, the measure of each interior angle is congruent to the other. 

 

A Theorem about Interior Angles

Here will prove the polygon interior angle sum theorem in the following paragraphs

Here’s the statement:

The sum of the interior angles of a polygon has n sides equals (2n – 4) × 900.

Here are the proofs:

In interior angles, the sum equals (2n – 4)

Here’s the proof:

A polygon ABCDE has n sides. Consider any point O within the polygon. OA, OB, OC are joined together.

 

When a polygon has “n” sides, it forms “n” triangles.

 

Summarizing the angles of a triangle yields a 180-degree angle, thus, n times 1800 is the sum of the angles of n triangles.

 

We can conclude that based on the above statement that Total angles in the interior + sum of angles in the interior = 2-n * 900 —-(1)

 

Nevertheless, the angles at O sum to 360 degrees

 

In (1), substituting the above value gives, 360 degrees + 2n * 90 degrees = total interior angles, therefore, the interior angles sum up to (2n × 900) – 3600, assuming that 90 is common, it becomes a product of (2n – 4) * 900 equals the sum of the interior angles. In other words, the sum of inner angles of “n” is (2n – 4) × 900.

 

Consequently, each interior angle of a regular polygon is ((2n – 4) × 900) / n

 

Regular polygons have the same measures for all interior angles.

 

However, in the 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:

 

 

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) 1800

30600 = (p – 2) 1800

p – 2 = 30600 / 1800

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) 1800

600 + 400 + (x + 83)0 = (3 – 2) 1800

1830 + x = 1800

x = 1800 – 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. 

 

The properties of Interior angles of a Polygon are important to learn. It prepares the base of concepts for geometric figures and students can proceed to understand advanced theories later with it.