How to Calculate the Sum of the Exterior Angles of a Polygon

By Carrie Perles
The exterior angles of an octogon each are 360/8 = 45 degrees.
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You can view the exterior angle of a polygon by extending one of the sides of a polygon and looking at the angle between the extension and its adjacent side. All polygons follow a rule that the sum of their exterior angles will equal 360 degrees. (Although you could draw two exterior angles at each of the polygon's vertices, this rule applies by taking the sum of only one exterior angle per vertex.) This rule is important in that it helps to determine other aspects of the polygon, such as the measurements of each external angle, each internal angle, and the number of sides the polygon has.

Regular Polygons

The angles of a regular polygon are equivalent, and their sides are as well. The sum of the exterior angles of a regular polygon will always equal 360 degrees. To find the value of a given exterior angle of a regular polygon, simply divide 360 by the number of sides or angles that the polygon has. For example, an eight-sided regular polygon, an octagon, has exterior angles that are 45 degrees each, because 360/8 = 45.

Irregular Polygons

The sum of the exterior angles of an irregular polygon also equals 360 degrees, even though the angles are not equivalent. Because irregular polygons have interior angles with different measurements, however, each exterior angle may have a different measurement as well. To find the measurement of an exterior angle, simply take the corresponding interior angle and subtract it from 180. Since the interior and exterior angle together add up to a straight line, their values should equal 180 degrees.

Checking the Values of Exterior Angles

In order to check that you have determined the right value for the exterior angles, you can add up all of the exterior angles for a given polygon to find their sum. If the sum is 360, you have identified all of the exterior angles correctly and accurately calculated their value.

Finding the Sides of a Regular Polygon from an Exterior Angle

If you know the value of an exterior angle of a regular polygon, you can easily find the number of sides that the polygon has as well. To do this, you would keep in mind that 360 divided by the number of sides of the polygon would result in the value of the exterior angle. Therefore, through the rule of cross multiplication, 360 divided by the value of one exterior angle would result in the number of sides of the polygon as well.

About the Author

Keren (Carrie) Perles is a freelance writer with professional experience in publishing since 2004. Perles has written, edited and developed curriculum for educational publishers. She writes online articles about various topics, mostly about education or parenting, and has been a mother, teacher and tutor for various ages. Perles holds a Bachelor of Arts in English communications from the University of Maryland, Baltimore County.