An exterior angle and an interior angle of a regular polygon are in the ratio 2 : 7. The number of sides in the polygon is :

1. 12

2. 6

3. 4

4. 9

Let n be the number of sides of the polygon.

Given,

An exterior angle and an interior angle of a regular polygon are in the ratio 2 : 7.

By formula,

Each interior angle of a regular polygon = $\dfrac{(2n - 4) × 90°}{n}$

Each exterior angle of a regular polygon = $\dfrac{360°}{n}$

$\Rightarrow \dfrac{\dfrac{360°}{n}}{\dfrac{(2n - 4) × 90°}{n}} = \dfrac{2}{7} \\[1em] \Rightarrow \dfrac{360° \times n}{(2n - 4) \times 90° \times n} = \dfrac{2}{7} \\[1em] \Rightarrow \dfrac{4}{2n - 4} = \dfrac{2}{7} \\[1em] \Rightarrow 2(2n - 4) = 28 \\[1em] \Rightarrow 4n - 8 = 28 \\[1em] \Rightarrow 4n = 28 + 8 \\[1em] \Rightarrow 4n = 36 \\[1em] \Rightarrow n = \dfrac{36}{4} = 9.$

Hence, Option 4 is the correct option.