Question

Each interior angle of a regular polygon lies between $136^\circ$ and $142^\circ$ , then the number of sides of the polygon is
A. $6$
B. $9$
C. $10$
D. $12$

Answer 1

Hint: We have given that each interior angle of a regular polygon lies between $136^\circ$ and $142^\circ$ . Here, we will use the formula of interior angles of a regular polygon. Then, using the conditions of the angles of a regular polygon we can find the sides of the polygon.
Formula used:
Interior angles of a Regular Polygon: $180^\circ - \dfrac{{360^\circ }}{n}$ , where $n$ is the sides of the polygon

Complete step by step solution:
We have given that each interior angle of a regular polygon lies between $136^\circ$ and $142^\circ$ . We need to check that in this case what can be the number of sides of the polygon.
Also, we know that,
Interior angles of a Regular Polygon: $180^\circ - \dfrac{{360^\circ }}{n}$ , where $n$ is the sides of the polygon
Let,
$\theta = 180^\circ - \dfrac{{360^\circ }}{n}$
And we know that $\theta$ lies between $136^\circ$ and $142^\circ$ .
We get this inequality
$136^\circ < \theta < 142^\circ$
$\Rightarrow 136^\circ < \;180^\circ - \dfrac{{360^\circ }}{n}\; < 142^\circ$ , as $\theta = 180^\circ - \dfrac{{360^\circ }}{n}$
We can also write it as
$136^\circ < \;180^\circ - \dfrac{{360^\circ }}{n}\;$ and $180^\circ - \dfrac{{360^\circ }}{n}\; < 142^\circ$
Let first take the inequality,
$136^\circ < \;180^\circ - \dfrac{{360^\circ }}{n}\;$
$\Rightarrow 136^\circ - 180^\circ < \; - \dfrac{{360^\circ }}{n}\;$
Simplifying of L.H.S., we get
$\Rightarrow - 44^\circ < \; - \dfrac{{360^\circ }}{n}\;$
$\Rightarrow 44^\circ > \dfrac{{360^\circ }}{n}\;$
By cross multiplying, we get
$\Rightarrow n > \dfrac{{360^\circ }}{{44^\circ }}\;$
$\Rightarrow n > 8.18$ - - - - - - $(1.)$
Now, let’s take the second inequality,
$180^\circ - \dfrac{{360^\circ }}{n}\; < 142^\circ$
$\Rightarrow - \dfrac{{360^\circ }}{n}\; < 142^\circ - 180^\circ$
Simplifying on R.H.S., we get
$\Rightarrow - \dfrac{{360^\circ }}{n}\; < - 38^\circ$
$\Rightarrow \dfrac{{360^\circ }}{n}\; > 38^\circ$
By cross multiplying, we get
$\Rightarrow 360^\circ \; > 38^\circ n$
$\Rightarrow 38^\circ n < 360^\circ$
Dividing by the coefficient of $n$ on both sides, we get
$\Rightarrow n < \dfrac{{360^\circ }}{{38^\circ }}$
$\Rightarrow n < 9.47$ - - - - - $(2.)$
From $(1.)$ and $(2.)$ , we get
$8.18 < n < 9.17$
Now, we know that $n$ is the number of sides, so $n$ is a natural number.
$n = 9$ , which satisfies the above inequality.

Thus, (B) is the correct option.

Note: In a regular polygon, all interior angles are of the same measure. But in the case of an irregular polygon, all the interior angles may have a different measure. Also, the sum of all the interior angles of a regular polygon varies according to the number of sides of the polygon.