The ratio between the interior angle and the exterior angle of a regular polygon is 2 : 1. Find.

(i) each exterior angle of the polygon,

(ii) number of sides in the polygon.

(i) It is given that the ratio between the interior angle and the exterior angle of a regular polygon is 2 : 1.

Let the common factor be a.

Then:

Interior angles = 5 x a = 2a

Exterior angles = 1 x a = a

We know that sum of the interior angle and the exterior angle is 180°.

⇒ 2a + a = 180°

⇒ 3a = 180°

⇒ a = $\dfrac{180°}{3}$

⇒ a = 60°

Thus:

Interior angle = 2a = 2 x 60° = 120°

Exterior angle = a = 60°

Hence, the exterior angle of the polygon is 60°.

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

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 120°

By cross-multiplying, we get

⇒ (2n - 4) x 90° = 120°n

⇒ 180°n - 360° = 120°n

⇒ 180°n - 120°n = 360°

⇒ 60°n = 360°

⇒ n = $\dfrac{360°}{60°}$

⇒ n = 6

Hence, the number of sides is 6.