The exterior angle of a regular polygon is one third of its interior angle. Find the number of sides in the polygon.

It is given that the exterior angle of a regular polygon is one-third of its interior angle.

Let the interior angle be a. Then, the exterior angle is $\dfrac{a}{3}$.

We know that the sum of the interior and exterior angles is 180°,

⇒ a + $\dfrac{a}{3}$ = 180°

⇒ $\dfrac{a}{1}$ + $\dfrac{a}{3}$ = 180°

⇒ $\dfrac{3a + a}{3}$ = 180°

⇒ $\dfrac{4a}{3}$ = 180°

⇒ a = $\dfrac{180° \times 3}{4}$

⇒ a = $\dfrac{540°}{4}$

⇒ a = 135°

So, the interior angle is 135°, and the exterior angle is:

$\dfrac{a}{3}$ = $\dfrac{135°}{3}$ = 45°

According to the properties of polygons, each interior angle of a regular polygon with n sides is $\dfrac{(2n - 4) \times 90°}{n}$.

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 135°

⇒ (2n - 4) x 90° = 135°n

⇒ 180°n - 360° = 135°n

⇒ 180°n - 135°n = 360°

⇒ 45°n = 360°

⇒ n = $\dfrac{360°}{45°}$

⇒ n = 8

Hence, the number of sides is 8.