The exterior angle of a polygon is that formed by one side of the figure and the prolongation of its continuous side. Thus, the angle is formed outside the polygon.
To understand it in another way, the exterior angle is one that shares the same vertex with an interior angle, being supplementary to it. That is, the exterior and interior angles of the same vertex add up to 180º or form a straight angle.
As we can see in the image above, the exterior angle of vertex D measures 56.3º, which corresponds to an interior angle of 123.7º.
The following equality can then be taken for granted, where x is the exterior angle and Ɵ is the interior angle of the respective vertex
Sum of exterior angles
The sum of the exterior angles of a polygon is equal to a complete angle, that is, 360º or 2π radians. This, regardless of the number of sides of the polygon.
We must specify that this calculation is taking into account only one external angle for each vertex. On the other hand, if we consider two, the total sum of the exterior angles of the polygon would be 720º or 4π radians.
That said, in the case of a regular polygon (where all the sides and interior angles measure the same), the exterior angle of all vertices are identical to each other and could be calculated with the following equation:
In the formula presented, x is the measure of the exterior angle and n, the number of sides of the regular polygon.
Exterior angle example
Suppose that the interior angle of a regular polygon is greater than its exterior angle by 90º. What shape is it and how large is its exterior angle?
First, we remember that the exterior and interior angle are supplementary. So if x is the exterior angle and Ɵ the interior angle:
Then, to know which polygon it is, we must remember that the sum of all exterior angles is 360º:
Therefore, we are facing a regular octagon.