Question

The measure of maximum possible exterior angle of a regular polygon is:
A) ${70^ \circ }$
B) ${60^ \circ }$
C) ${90^ \circ }$
D) ${120^ \circ }$

Answer 1

Hint:
We will use the formula to calculate the measure of exterior angle of a regular polygon given by: ${\text{exterior angle = 18}}{{\text{0}}^ \circ } - {\text{ interior angle}}$. So, for maximum exterior angle, the measure of interior angle must be minimum. Hence, we will check all regular polygons for minimum interior angle using the definition of the regular polygon.

Complete step by step solution:
We are required to calculate the maximum possible exterior angle of a regular polygon.
For this, we know that exterior angle of any polygon is calculated using interior angle of the polygon by the formula: ${\text{exterior angle = 18}}{{\text{0}}^ \circ } - {\text{ interior angle}}$
Now, for the maximum value of exterior angle, the interior angle of the polygon must be minimum.
Since we are considering regular polygons, a regular polygon is defined as a polygon which is equilateral (having all sides of same length) and equiangular (all angles of the polygon are equal).
So, if we see for various regular polygons such as equilateral triangle, square, regular pentagon, regular hexagon, etc., the interior angles are of measure \[{60^ \circ }\], ${90^ \circ }$, ${108^ \circ }$, ${120^ \circ }$, etc. respectively.
We can deduce that the minimum possible interior angle will be present in an equilateral triangle (i.e., interior angle = ${60^ \circ }$) as measure of interior angles increase with the number of sides of a regular polygon.
Using the formula: ${\text{exterior angle = 18}}{{\text{0}}^ \circ } - {\text{ interior angle}}$, we can calculate the maximum possible exterior angle of a regular polygon (equilateral triangle):
$
   \Rightarrow {\text{ Exterior angle = 18}}{{\text{0}}^ \circ } - {60^ \circ } \\
   \Rightarrow {\text{ Exterior angle = }}{120^ \circ } \\
 $

So, the maximum possible exterior angle of a regular polygon is ${120^ \circ }$ .
Hence, option (D) is correct.

Note:
In this question, you may get confused in the formula used i.e., ${\text{exterior angle = 18}}{{\text{0}}^ \circ } - {\text{ interior angle}}$, as we know that exterior angle of a polygon is formed by any side of a polygon and extending its adjacent side. So, they form a straight line and the measure of angle of a straight line is \[{180^ \circ }\]. You can also calculate the measure of exterior angle of regular polygons using the formula: $\dfrac{{{{360}^ \circ }}}{n}$, where n is the number of sides of the regular polygon and then by comparing, you can check the options.