The exterior angles of a hexagon are the angles formed when we extend the sides of the hexagon. These angles have a total sum of 360°. If the hexagon is regular, we can simply divide the sum by 6 to get the measure of each exterior angle. However, to determine the exterior angle measures of an irregular hexagon, we need to use other methods.
Here, we will look at some examples of exterior angles of hexagons.
Exterior angles in a regular hexagon
A regular hexagon is characterized by having all six sides with the same length and all its internal angles with the same measure. Furthermore, these hexagons also have all their exterior angles with the same measure.
We can determine the measure of each exterior angle in a regular hexagon by considering that the sum of the exterior angles of any polygon is always equal to 360°. Therefore, we simply divide by 6 to get each angle:
360°÷6 = 60°
Each exterior angle measures 60°.
Formula for finding exterior angles in a hexagon
We can calculate the measure of the exterior angles in a hexagon considering that the sum of an exterior angle and its corresponding interior angle is equal to 180°. Therefore, if we have an interior angle x in a hexagon, the exterior angle is equal to:
In addition, we can also determine exterior angle measures by using the sum total of 360° and subtracting the known angle measures.
How to find a missing exterior angle in a hexagon?
In a regular hexagon, the measures of the exterior angles are always equal, so their measures are always 60° each. However, in the case of irregular hexagons, the measurements of the exterior angles are different.
To calculate the measures of a missing exterior angle in a hexagon, we have to add the measures of the known angles and subtract the result from 360°.
For example, the following hexagon has exterior angles 50°, 60°, 70°, 80°, 65°.
Let’s add the measures of the known angles:
50°+60°+70°+80°+65° = 325°
Therefore, we find the measure of the sixth angle by subtracting the result from 360°:
360°-325° = 35°
It is also possible to find the measures of the exterior angles if we have the measures of the interior angles. Therefore, if we have the measure of an interior angle, we subtract it from 180° to get the measure of the corresponding exterior angle.
For example, in the following hexagon, we have the interior angles 120°, 140°, 110°, 100°, 120°, 130°.
We subtract each 180° angle to get the corresponding exterior angle:
The exterior angle measures are 60°, 30°, 70°, 80°, 60°, and 50°.
Solved examples of exterior angles of hexagons
Find the missing exterior angle measures in the hexagon below.
Solution: The angles marked with a double line and that have the same color share the same measure, so we have a=50°. To find the measure of angle b, we have to add the known angle measures and subtract the result from 360°. Therefore, we have:
50°+50°+70°+75°+60° = 305°
⇒ 360°-305° = 55°
The measure of angle b is 55°.
Determine the measures of the exterior angles of the hexagon below.
Solution: We can find the measure of each exterior angle by subtracting the measure of the corresponding interior angle from 180°:
Now, we can find the missing angle by adding the known angles and subtracting from 360°:
Interested in learning more about the interior angles of a polygon? Take a look at these pages: