# Each interior angle of a regular polygon lies between 136 to 142. How do we calculate the sides of the polygon?

Jan 7, 2016

$n = 9$

#### Explanation:

In a regular polygon each interior angle can be obtained in this way:
$\alpha = {180}^{\circ} - {360}^{\circ} / n$

From the conditions of the problem:
${136}^{\circ} < \alpha < {142}^{\circ}$

That's the conjugation of this two inequations:
${136}^{\circ} < \alpha$ and $\alpha < {142}^{\circ}$

Resolving the first inequation
${136}^{\circ} < {180}^{\circ} - {360}^{\circ} / n$
${136}^{\circ} < \frac{{180}^{\circ} \cdot n - {360}^{\circ}}{n}$ => ${136}^{\circ} \cdot n < {180}^{\circ} . n - {360}^{\circ}$ => ${44}^{\circ} . n > {360}^{\circ}$ => $n > 8.18$

Resolving the second inequation
${180}^{\circ} - {360}^{\circ} / n < {142}^{\circ}$
${180}^{\circ} \cdot n - {360}^{\circ} < {142}^{\circ} \cdot n$ => ${38}^{\circ} . n < {360}^{\circ}$ => $n < 9.47$

Conjugating the two inequations
$8.18 < n < 9.47$

Since $n \in \mathbb{N}$, its only value that satisfies the inequation is $n = 9$

By the way
$\alpha = {180}^{\circ} - {360}^{\circ} / 9 = {180}^{\circ} - {40}^{\circ}$ => $\alpha = {140}^{\circ}$