Solved Examples on Interior Angles Formula

Example 1: Determine the sum of interior angles of a regular heptagon.

Solution: 

We know that a heptagon has seven sides, i.e., n = 7.

From the sum of the interior angles of a given polygon,

We have, S = (n − 2) × 180°

= (7 − 2) × 180°

= 5 × 180° = 900°

Hence, the sum of interior angles of a regular heptagon is 900°.

Example 2: Find the interior angle at vertex C in the figure given below.

Solution:

The given polygon is a pentagon. We know that a pentagon has seven sides, i.e., n = 5.

From the sum of the interior angles of a given polygon,

We have, S = (n − 2) × 180°

= (5 − 2) × 180° = 540°

Now, we have the sum of all the interior angles in the given polygon is 540°.

∠A + ∠B + ∠C + ∠D + ∠E = 540°

x + 111° + (x – 20°) + 105° + (x – 10°) = 540°

3x + 186° = 540°

3x = 354°

x = 118°

Now, let the angle at ∠C = x – 20°

= 118° – 20°

∠C = 98°

Thus, the interior angle at vertex C is ∠C = 98°.

Example 3: What is the sum of the interior angles of a polygon having 12 sides?

Solution: 

Given data:

The number of sides of the given polygon (n) = 12

From the sum of the interior angles of a given polygon,

We have, S = (n − 2) × 180°

= (12 − 2) × 180°

= 10 × 180° = 1800°

The sum of interior angles of a polygon having 12 sides is equal to 1800°.

Example 4: What is the measure of the interior angles of a regular polygon having 8 sides?

Solution: 

Given data:

The number of sides of the given polygon (n) = 8

Since the given polygon is regular, all the angles are equal.

We have,

The measure of each interior angle of a regular polygon = (n – 2) × 180°/n

= [(8 − 2) × 180°]/8

= 1,080°/8 = 135°

Hence, the measure of each interior angle of a regular polygon having 8 sides is 135°.

Example 5: Find the interior angle at vertex F in the figure given below.

Solution:

The given polygon is a pentagon. We know that a pentagon has seven sides, i.e., n = 5.

From the sum of the interior angles of a given polygon,

We have, S = (n − 2) × 180°

= (6 − 2) × 180° = 720°

Now, we have the sum of all the interior angles in the given polygon is 720°.

∠A + ∠B + ∠C + ∠D + ∠E + ∠F = 720°

132° + y° + (y – 15)° + 110° + (y – 20)° + 99° = 720°

3y + 306° = 720°

3y = 414°

y = 138°

Now, let the angle at ∠F = y – 20°

= 138° – 20°

∠F = 118°

Thus, the interior angle at vertex F is ∠F = 118°.

Polygon is defined as a two-dimensional geometric figure that has a finite number of line segments connected to form a closed shape. The line segments of a polygon are called edges or sides, and the point of intersection of two edges is called a vertex. The angle of a polygon is referred to as the space formed at the intersection point (vertex) of two adjacent sides.

A polygon is of two types: a regular polygon and an irregular polygon. A regular polygon is a polygon whose all sides and all interior angles are measured the same, whereas an irregular polygon is a polygon whose all sides and all interior angles do not measure the same. And we also have different types of polygons like triangles, quadrilaterals, pentagons, hexagons, etc, based on the number of sides of a polygon. Every polygon has interior angles and exterior angles, where an interior angle is the one that lies inside the polygon and the exterior angle is the one that lies outside the polygon.