Calculating Degrees of An Exterior Angle

An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side. Calculating the degrees of an exterior angle is essential in geometry, construction, and design. This guide provides a comprehensive explanation of how to calculate exterior angles, including the formula, practical examples, and an interactive calculator.

What is an Exterior Angle?

An exterior angle is formed when you extend one side of a polygon and measure the angle between this extension and the adjacent side. Exterior angles are particularly important in geometry because they help determine the properties of polygons and can be used to calculate the sum of all exterior angles of a polygon.

For any polygon, the sum of all exterior angles is always 360 degrees, regardless of the number of sides. This property is known as the Exterior Angle Sum Theorem.

How to Calculate Exterior Angle Degrees

Calculating the degrees of an exterior angle involves understanding the relationship between the interior and exterior angles of a polygon. Here's a step-by-step guide:

Identify the interior angle of the polygon. This is the angle formed by two adjacent sides inside the polygon.
Use the formula for calculating the exterior angle, which is: Exterior Angle = 180° - Interior Angle.
If you know the number of sides of the polygon and the sum of the interior angles, you can calculate the measure of each interior angle and then find the exterior angle.

Important Note

For regular polygons (where all sides and angles are equal), the measure of each exterior angle can be calculated by dividing 360 degrees by the number of sides.

The Exterior Angle Formula

The formula for calculating the degrees of an exterior angle is straightforward:

Exterior Angle Formula

Exterior Angle = 180° - Interior Angle

This formula works for any polygon, whether it's a triangle, quadrilateral, pentagon, or any other polygon. The interior angle is the angle formed by two adjacent sides inside the polygon, and the exterior angle is the angle formed by one side and the extension of an adjacent side.

Practical Examples

Let's look at some practical examples to illustrate how to calculate exterior angles.

Example 1: Triangle

Consider a triangle with an interior angle of 60 degrees. To find the exterior angle:

Calculation

Exterior Angle = 180° - 60° = 120°

The exterior angle for this triangle is 120 degrees.

Example 2: Quadrilateral

For a quadrilateral with an interior angle of 100 degrees, the exterior angle is:

Calculation

Exterior Angle = 180° - 100° = 80°

The exterior angle for this quadrilateral is 80 degrees.

FAQ

What is the sum of all exterior angles of a polygon?: The sum of all exterior angles of any polygon is always 360 degrees, regardless of the number of sides.
How do you calculate the exterior angle of a regular polygon?: For a regular polygon, divide 360 degrees by the number of sides to find the measure of each exterior angle.
Can exterior angles be greater than 180 degrees?: Yes, exterior angles can be greater than 180 degrees if the polygon is concave.
What is the relationship between interior and exterior angles?: The interior and exterior angles are supplementary, meaning they add up to 180 degrees.
How are exterior angles used in real-world applications?: Exterior angles are used in construction, architecture, and design to determine the angles at which walls and structures meet.