Find the number of sides in a regular polygon, if its each exterior angle is:

(i) $\dfrac{1}{3}$ of a right angle

(ii) two-fifths of a right angle

(i) $\dfrac{1}{3}$ of a right angle

= $\dfrac{1}{3}$ x 90°

= $\dfrac{90°}{3}$

= 30°

According to the properties of a polygon, if there are n sides, then each of its exterior angles is $\dfrac{360°}{n}$.

Given that each exterior angle is 30°,

⇒ $\dfrac{360°}{n}$ = 30°

⇒ n = $\dfrac{360°}{30°}$

⇒ n = 12

Hence, the number of sides is 12.

(ii) Two-fifths of a right angle,

= $\dfrac{2}{5}$ x 90°

= $\dfrac{180°}{5}$

= 36°

According to the properties of a polygon, if there are n sides, then each of its exterior angles is $\dfrac{360°}{n}$.

Given that each exterior angle is 36°,

⇒ $\dfrac{360°}{n}$ = 36°

⇒ n = $\dfrac{360°}{36°}$

⇒ n = 10

Hence, the number of sides is 10.