Calculate the number of sides of a regular polygon if:

(i) its interior angle is five times its exterior angle.

(ii) the ratio between its exterior angle and interior angle is 2 : 7.

(iii) its exterior angle exceeds its interior angle by 60°.

(i) It is given that the interior angle of a polygon is five times its exterior angle.

Let the exterior angle be a. Then, the interior angle is 5a.

Since the sum of the interior angle and exterior angle is 180°.

⇒ 5a + a = 180°

⇒ 6a = 180°

⇒ a = $\dfrac{180°}{6}$

⇒ a = 30°

Thus, the exterior angle is 30° and the interior angle is 5a = 5 x 30° = 150°.

According to the properties of polygons, if a polygon has n sides, then each of its interior angles is $\dfrac{(2n - 4) \times 90°}{n}$.

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 150°

⇒ (2n - 4) x 90° = 150°n

⇒ 180°n - 360° = 150°n

⇒ 180°n - 150°n = 360°

⇒ 30°n = 360°

⇒ n = $\dfrac{360°}{30°}$

⇒ n = 12

Hence, the number of sides is 12.

(ii) It is given that the ratio between the exterior angle and the interior angle is 2 : 7.

Let the exterior angle be 2a. Thus, the interior angle is 7a.

Since the sum of the interior angle and exterior angle is 180°.

⇒ 2a + 7a = 180°

⇒ 9a = 180°

⇒ a = $\dfrac{180°}{9}$

⇒ a = 20°

Thus, the exterior angle is 2a = 2 x 20° = 40° and the interior angle is 7a = 7 x 20° = 140°.

According to the properties of polygons, if a polygon has n sides, then each of its interior angles is $\dfrac{(2n - 4) \times 90°}{n}$.

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 140°

⇒ (2n - 4) x 90° = 140°n

⇒ 180°n - 360° = 140°n

⇒ 180°n - 140°n = 360°

⇒ 40°n = 360°

⇒ n = $\dfrac{360°}{40°}$

⇒ n = 9

Hence, the number of sides is 9.

(iii) It is given that the exterior angle of a polygon exceeds its interior angle by 60°.

Let the interior angle be a. Therefore, the exterior angle = 60° + a.

Since the sum of the interior angle and the exterior angle is 180°.

⇒ a + (60° + a) = 180°

⇒ a + 60° + a = 180°

⇒ 2a + 60° = 180°

⇒ 2a = 180° - 60°

⇒ 2a = 120°

⇒ a = $\dfrac{120°}{2}$

⇒ a = 60°

Thus, the interior angle is 60° and the exterior angle is:

a + 60° = 60° + 60° = 120°

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

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 60°

⇒ (2n - 4) x 90° = 60°n

⇒ 180°n - 360° = 60°n

⇒ 180°n - 60°n = 360°

⇒ 120°n = 360°

⇒ n = $\dfrac{360°}{120°}$

⇒ n = 3

Hence, the number of sides is 3.