Question

(i) Is it possible to have a regular polygon with a measure of each exterior angle ‘a’ as 22°?
(ii) Can it be an interior angle of a regular polygon? Why?

Answer 1

(i) Since, the sum of all the exterior angles of a regular polygon = 360°, which is not divisible by 22°.
It is not possible for a regular polygon to have its exterior angle as 22°.

(ii) Sum of all interior angles of a regular polygon = (n-2) $\times {180}^{\circ }$
Measure of its each angle = $\frac{(n-2)\times {180}^{\circ }}{n}$
So,$\frac{(n-2)\times {180}^{\circ }}{n}={22}^{\circ }\Rightarrow 80n-(2\times 180)=22n\Rightarrow 180n-360=22n\Rightarrow 158n=360\Rightarrow n=\frac{360}{158}=\frac{180}{79}$
Here, n is not a whole number.
Since, the number of sides cannot be in fractions,
It is not possible for a regular polygon to have its interior angle = ${22}^{\circ }$.