# Sum of Angles Formula

We use the sum of angles formula to determine the sum of interior angles of a polygon. The sum of angles in a polygon depends on the number of vertices it has. When there is a polygon with four or more than four sides, we draw all the possible diagonals from one vertex. Then the polygon is broken into several non-overlapping triangles. Let us learn about the sum of angles formula with a few examples in the end.

## What Is the Sum of Angles Formula?

The formula for interior angles can be determined by multiplying the number of triangles by 180°and the total number of triangles is two less than the number of sides of a polygon, always. The sum of angles formula of a given polygon can be expressed as

- The sum of the interior angles of a given polygon = (n − 2) × 180°, where n = the number of sides of the polygon.
- The sum of exterior angles of a given polygon = 360°

(n-2) denotes the number of triangles that a polygon could be divided into and 180° is for counting the number of degrees in a triangle.

Let us have a look at a few solved examples on the sum of angles in a polygon to understand the concept better.

## Example on Sum of Angles Formula

**Example 1:** George cuts a piece of paper into a regular pentagonal polygon and he wants to know the sum of interior angles of the regular pentagon. Find the sum of interior angles of a regular pentagon for George.

**Solution:**

To find: The sum of interior angles of a regular pentagon.

Sides of pentagon (n) = 5 (given)

Using sum of angles formula,

The sum of its interior angle of a given polygon (S) = ( n − 2) × 180°

S = ( 5 − 2) × 180°

= (3) × 180°

= 540°

**Answer: **The sum of the interior angles of a regular pentagon is 540°

**Example 2:** Using the angle sum formula verify that sum of interior angles of a triangle is 180 degrees.

**Solution: **To verify the sum of interior angles of a triangle is 180 degrees.

Angle sum formula = ( n − 2) × 180°.

Sides of a triangle = 3

Putting the value of n = 2 in angle sum formula we have, (3 − 2) × 180° = 180°

**Answer:** Hence proved the sum of interior angles of a triangle is 180 degrees.

**Example 3: **Harry wants to find the interior angle of the hexagon brick. Help him to find the measure of each interior angle of a regular hexagon.

**Solution:** To find: The measure of each interior angle of a regular hexagon

Sides of Hexagon (n) = 6 (given)

Using the sum of angles formula,

The sum of the interior angles of a given polygon (S) = ( n − 2) × 180°

S = ( 6 − 2) × 180°

= (4) × 180°

S = 720°

The measure of each interior angle of a regular hexagon = 720°/6 = 120°

**Answer:** The measure of each interior angle of a regular hexagon brick is 120°

## FAQ's on Sum of Angles Formula

### What Does n Represent in the Sum of Angles Formula?

The sum of angles formula is given as ( n − 2) × 180°. Here n denotes the number of sides of a polygon.

### How To Find the Sum of Angles of a Rectangle Using Sum of Angles Formula?

To find the sum of interior angles of a rectangle we can directly use the sum of angles formula. By putting the value of n = 4 we can easily get the answer.

Sum of interior angles of a rectangle = ( n − 2) × 180° = (4 - 2) × 180° = 2 × 180° = 360°.

### How To Find the Sum of Angles of a Square Using Sum of Angles Formula?

To find the sum of interior angles of a square we can directly use the sum of angles formula. By putting the value of n = 4.

Sum of interior angles of a square = ( n − 2) × 180° = (4 - 2) × 180° = 2 × 180° = 360°.

### How To Find the Sum of Angles of a Decagon Using Sum of Angles Formula?

To find the sum of interior angles of a decagon we can directly use the sum of angles formula. By putting the value of n = 10 we can easily get the answer.

Sum of interior angles of a decagon = (10 − 2) × 180° = (8) × 180° = 1440°.

visual curriculum