Exterior Angles of a Pentagon – Formula and Examples

The exterior angles of a pentagon are formed when we extend the sides of the pentagon. These angles have a total sum of 360°. Also, the sum of an exterior angle and its corresponding interior angle is equal to 180°. Therefore, we can use these properties to find the different measures of the exterior angles of pentagons.

Here, we will learn more details about these angles and we will solve some practice examples.

GEOMETRY
exterior angles in a pentagon 1

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Learning about the exterior angles of a pentagon with examples.

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GEOMETRY
exterior angles in a pentagon 1

Relevant for

Learning about the exterior angles of a pentagon with examples.

See angles

Exterior angles in a regular pentagon

Regular pentagons are polygons with five sides, which have all their sides with the same length and all their interior angles with the same measure. This also means that all exterior angles in a regular pentagon are equal.

Since the sum of exterior angles in any polygon is always equal to 360°, we can find the measure of each exterior angle of a regular pentagon by dividing 360° by 5. Therefore, we have:

360°÷5 = 72°

Each exterior angle measures 72°.

exterior angles in a pentagon

Formula to find the exterior angles of a pentagon

Exterior angles can be found by considering that the sum of an exterior angle and its corresponding interior angle is equal to 180°. Therefore, if x is the interior angle, the exterior angle is equal to:

180°-x

Additionally, we can also use the sum total of 360° to find the exterior angle measures.


How to find a missing exterior angle in a pentagon?

If the pentagon is regular, we know that all exterior angles measure 72°. However, when a pentagon is irregular, the exterior angle measures are different. Therefore, if we know the measures of four angles, we can find the measure of the fifth angle by subtracting the sum of the known angles from 360°.

For example, in the following pentagon, we have the angles 60°, 50°, 80°, 100°.

example 1 of exterior angles in a pentagon

Let’s add to the four angles given:

60°+50°+80°+100° = 290°

Now, we subtract this value from 360° to find the measure of the fifth angle:

360°-290° = 70°

Additionally, we can also use the interior angle measures of a pentagon to calculate the corresponding exterior angle measures.

For example, in the following pentagon, we have interior angles 100°, 110°, 80°, 90°, 160°.

example 2 of exterior angles in a pentagon

We can obtain the exterior angle measures by subtracting each interior angle from 180°:

180°-100° = 80°

180°-110° = 70°

180°-80° = 100°

180°-90° = 90°

180°-160° = 20°

The exterior angle measures are 80°, 70°, 100°, 90°, and 20°.


Solved examples of exterior angles of a pentagon

EXAMPLE 1

Determine the measures of the exterior angles of the pentagon.

example 3 of exterior angles in a pentagon

Solution: The angles that are marked with a double line and have the same color are the same, so we have a=60°. To find the measure of the missing angle, we add the measures of the known angles and subtract the result from 360°:

60°+60°+80°+90° = 290°

⇒  360°-290° = 70°

The measure of angle b is 70°.

EXAMPLE 2

Find the measures of the exterior angles of the pentagon.

example 4 of exterior angles in a pentagon

Solution: We have to subtract each corresponding interior angle from 180° to find the exterior angle measures. Therefore, we have:

180°-110° = 70°

180°-120° = 60°

180°-100° = 80°

180°-90° = 90°

Now, we have a missing angle. We can find that angle by adding the known angles and subtracting from 360°:

70°+60°+80°+90° = 300°

360°-300° = 60°


See also

Interested in learning more about angles of a pentagon and other polygons? Take a look at these pages:

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