Question

What is the sum of the exterior angles of a 30-gon?

Answer 1

Hint: This type of question depends on the concept of exterior angles of a polygon. Exterior angle is an angle at a vertex of a polygon, present outside the polygon, formed by the extension of its adjacent sides. Exterior angles of a regular polygon are equal in measure.

Complete step by step answer:
Now, as we have to find out the sum of the exterior angles of a 30-gon so we consider the following:
Let us start from the vertex at angle 1. Now we travel in an anti-clockwise direction, through angles 2, 3, 4, 5, and so on up-to angle 30 and coming back to the same vertex. Actually we covered the entire perimeter of the polygon and hence, made one complete turn in the process. As all of us know, one complete turn corresponds to \[{{360}^{\circ }}\]. Thus, we can say that the \[\angle 1,\angle 2,\angle 3,\angle 4,..........\angle 30\] sum up-to \[{{360}^{\circ }}\].

Hence, the sum of the exterior angles of a 30-gon is \[{{360}^{\circ }}\].

Note: In this type of question students may use the concept of linear pair and interior angle to find the sum of exterior angles. We know that, for a polygon with n sides that is n-gon the sum of the exterior angles is equal to the difference between the sum of the linear pairs and the sum of the interior angles.
\[\begin{align}
  & \Rightarrow \text{Sum of exterior angles = 18}{{0}^{\circ }}n-\text{18}{{0}^{\circ }}\left( n-2 \right) \\
 & \Rightarrow \text{Sum of exterior angles = 18}{{0}^{\circ }}n-\text{18}{{0}^{\circ }}n+{{360}^{\circ }} \\
 & \Rightarrow \text{Sum of exterior angles = }{{360}^{\circ }} \\
\end{align}\]
Hence, for any polynomial with n sides that is n-gon, the sum of exterior angles is \[{{360}^{\circ }}\].
Thus, the sum of the exterior angles of a 30-gon is \[{{360}^{\circ }}\].