Question

The measurement of each angle of a polygon is \[{160^ \circ }\]. The number of its sides is?
A) 15
B) 18
C) 20
D) 30

Answer 1

Hint: In this question first we will assume the sides of a polygon be n, then we will substitute the given value in the formula of an interior angle of a polygon. On simplifying that we will get the required answer.

Complete step by step solution: We have been given the measurement of each angle of a polygon is \[{160^ \circ }\].
Let us suppose there be n sides, in the polygon.
To find the value of n using the given data we will use the formula to find the measure of each interior angle of a polygon which is \[\dfrac{{\left( {n - 2} \right)180}}{n};\] where n is the number of sides of the polygon.
Here, we have given the measurement of each angle of a polygon is\[{160^ \circ }\]so, now we equate it with \[\dfrac{{\left( {n - 2} \right)180}}{n}\], we get,
\[160 = \dfrac{{\left( {n - 2} \right)180}}{n}\]
\[ \Rightarrow 160n = \left( {n - 2} \right)180\]
\[ \Rightarrow 160n = 180n - 360\]
\[ \Rightarrow 360 = 180n - 160n\]
Taking common n from the right-hand side we get,
\[360 = n\left( {180 - 160} \right)\]
\[ \Rightarrow 360 = n\left( {20} \right)\]
\[ \Rightarrow n = \dfrac{{360}}{{20}}\]
\[ \Rightarrow n = 18\]
As we have assumed n is the total number of sides of a polygon it implies that the polygon has 18 sides.

Hence, option B. 18 is the correct answer.

Note: There is another approach to solve the above question which is mentioned below:
We have been given the measure of each angle of a polygon is \[{160^ \circ }\]
Let us assume n be the number of sides of a polygon.
Therefore, using the formula: exterior angle \[ = {180^ \circ } - \] interior angle
Now, put the value of the interior angle in the above formula we get,
Exterior angle \[ = {180^ \circ } - {160^ \circ }\]
\[ \Rightarrow \] Exterior angle \[ = {20^ \circ }\]
Which implies that the measure of each exterior angle of the given polygon is \[{20^ \circ }\]
There is a relationship between the number of sides of a polygon and the exterior angle of a polygon which is given below:
The measure of each exterior angle \[ = \dfrac{{{{360}^ \circ }}}{n}\]
\[ \Rightarrow {20^ \circ } = \dfrac{{{{360}^ \circ }}}{n}\]
\[ \Rightarrow n = \dfrac{{{{360}^ \circ }}}{{{{20}^ \circ }}}\]
\[ \Rightarrow n = 18\]
Therefore, the number of sides of the polygon are 18.