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

The exterior angles of an octogon each are 360/8 = 45 degrees.
••• Mega_Pixel/iStock/Getty Images

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.

Related Articles

How to Find an Angle of a Hexagon
How to Find the Number of Sides of a Polygon
How To: Degree to Radian Conversion
How to Solve a Hexagon
How to Find Degrees in Polygons
How to Find Angles & Sides of a Triangle
How to Calculate the Obtuse Triangle
How to Figure Out Complementary And Supplementary Angles
How to Calculate Sides of a Triangle
Facts About Parallelograms
How to Calculate Hypotenuse
How to Find Angles in a Trapezoid
How to Calculate the Apothem of a Polygon
How to Find the Area of a Scalene Triangle
How to Calculate an Angle With Trig
How to Calculate Triangle Dimensions
How to Use Trigonometry in Carpentry
How to Find Angle Measures in a Quadrilateral
How to Find a Missing Angle
What Are Congruent Shapes?