In a regular polygon, the ratio of an interior angle : exterior angle = 14 : 1.

1. Use the given ratio to express the interior and exterior angles.

Let the interior angle be $14x$ and the exterior angle be $x$.

2. Find the size of one exterior angle.

An interior angle and its exterior angle sum to $180^\circ$.

$14x+x=180$
$15x=180$
$x=12$

So each exterior angle is $12^\circ$.

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

For a regular polygon,

$\text{number of sides}=\frac{360^\circ}{\text{exterior angle}}$

$n=\frac{360}{12}=30$

1. Find the sum of the interior angles using $n=30$.

From part (a), the polygon has $30$ sides.

Sum of interior angles
$=(n-2)\times 180^\circ$
$=(30-2)\times 180^\circ$
$=28\times 180^\circ=5040^\circ$