For a regular polygon with n sides, the ratio of the size of an interior angle to its exterior angle is 7 : 2.

1. Use the ratio of interior angle to exterior angle to find the interior angle.

Interior angle : exterior angle $= 7 : 2$

Total parts $= 7+2=9$.

Since an interior angle and its exterior angle add up to $180^\circ$,

$\text{interior angle}=\frac{7}{9}\times 180^\circ=140^\circ.$

1. Find the exterior angle using the interior angle from part (a).

From part (a), the interior angle is $140^\circ$.

Since the interior angle and exterior angle of a polygon add to $180^\circ$,

$\text{exterior angle} = 180^\circ - 140^\circ = 40^\circ$.

2. Use the exterior angle of a regular polygon to find the number of sides.

For a regular polygon,

$\text{exterior angle} = \dfrac{360^\circ}{n}$.

So

$\dfrac{360}{n} = 40$

$n = \dfrac{360}{40} = 9$.