Question

Is it possible to have a regular polygon with measure of each exterior angle as $22°$? Can it be an interior angle of a regular polygon? Why?

Answer 1

Step $1$: Compute the total number of sides of the polygon if the exterior angle is $22°$.

Given: A regular polygon with each exterior angle is $22°$.

we know that, (number of sides)$\times$(a measure of the exterior angle)$=360°$

$\Rightarrow$ (number of sides)$\times$$22°=360°$

$\Rightarrow$ number of sides$=\frac{360°}{22°}$

$\Rightarrow$ number of sides$=16.36$

The number of sides can't be a decimal value. It should be an integer value.

Hence, it is not possible to have a regular polygon with the measure of each exterior angle as $22°$.

Step $2$: Compute the total number of sides of the polygon if the interior angle is $22°$.

Given: A regular polygon with each interior angle is $22°$.

Each exterior angle of the polygon $=180°-22°$

Each exterior angle of the polygon $=158°$

we know that,

(number of sides)$\times$(measure of the exterior angle)$=360°$

$\Rightarrow$ (number of sides)$\times$$158°=360°$

$\Rightarrow$ number of sides $=\frac{360°}{158°}$

$\Rightarrow$ number of sides $=2.28$

The number of sides can't be a decimal value it should be an integer value.

Hence, it is not possible to have a regular polygon with the measure of each interior angle as $22°$.

Hence, it is not possible to have a regular polygon with measure of each exterior angle as $22°$ or interior angle as $22°$.