When the polygons are formed, and one of its sides is extended longer than the vertex of a corner, the exterior angle of the polygon is formed. The result of the sum of the exterior angles of a polygon is 360 degrees.
On a side note, we can use this piece of information in the exterior angle of a polygon formula to solve various questions.
It is presumed that we all know what a polygon is and its characteristic features for recapitulation. The sides are made up of the line segments, and the vertex is made up of the pair of sides which meet at a point. That meeting point is called the vertex.
The interior angle is one of the vertices of the polygon. The interior and exterior angles of a polygon are different for different types of polygons. Now it is time to take a closer look at the exterior angles and study the concept of exterior angles of a polygon.
Properties Of Exterior Angles Of a Polygon
The exterior angle is formed on one of the sides of a closed figure which has its extension in the means of length with respect to its adjacent side. The figure given below is a pentagon consisting of 5 sides as well as five vertices. We will look into what is the exterior angle of a regular pentagon as we move ahead and also look into its different aspects.
The extension of its sides has caused the exterior angles of the pentagon with respect to its adjacent sides. Here we will be talking about the exterior angle of a regular pentagon with five vertices, and at each side, each angle is to be considered.
They are formed either on the outside or exterior of the polygon.
The summation of an interior angle and its following exterior angle Is every time 180 degrees because they lie on the same straight line on the same plane.
In the figure, angles 1, 2, 3, 4, and 5 are to be noted as the exterior angles of the polygon. The exterior angles of a regular polygon are always equivalent in measurement.
Sum Of Exterior Angles Of A Polygon
If we start traversing from the vertex at angle 1 and continue to see in a clockwise direction, we have to bend at the points or vertexes at the angle 2,3,4,5 and then return to the same vertex. Now, we have travelled the whole boundary of the polygon and have made a full turn in the procedure. This full turn is equivalent to 360 degrees. Therefore, it can be concluded that angle 1,2,3,4 and 5 totally sum up to 360 degrees.
So, we can summarise that the sum of the exterior angles of a polygon is equal to 360 degrees without considering the number of sides of the polygon. The total of the exterior angles of the polygon does not depend on the number of sides of the polygon.
Polygon Exterior Angle Sum Theorem
If we consider that a polygon is a convex polygon, the summation of its exterior angles at each vertex is equal to 360 degrees.
Now it’s the time where we should see the sum of exterior angles of a polygon proof.
Proof: Let us Consider a polygon with m number of sides or an m-gon. The sum of its exterior angles is M. By the exterior angle of a polygon theorem,
For any enclosed structure, formed by sides and vertex, the summation of the exterior angles is always equivalent to the summation of linear pairs and sum of interior angles. Hence,
M= 180m – 180(m-2)
Now, after framing the equation, we should solve it step by step. First, simple algebraic operations have to be completed. Then the like algebraic term in geometry needs to be put down together in a group, and the required basic algebraic operations need to be done. Attention needs to be paid here to see whether any like term in the groupings gets cancelled or not.
M = 180m – 180m + 360
Now, after this step, it becomes more compact and easy to complete the sum as the grouping has been done, and the required basic algebraic operations need to be completed.
M = 360 degrees
So here, we got the addition of exterior angles of m vertices equivalent to 360 degrees.
Identify the type of regular polygon whose exterior angle measures 60 degrees.
Since the polygon is regular, the measure of all the interior angles needs to be the same. Hence, all its exterior angles are to be measured in the same as well, i.e., 60 degrees.
As the sum of the exterior angle of a polygon is 360 degrees and each one measures 60 degrees, we
Number of angles = 360/60 = 6
Since the polygon has 6 exterior angles, it has 6 sides. Hence it is a hexagon.