The ratio of the interior angle to the exterior angle of a regular polygon is $5:2$. We need to find the number of sides of the polygon.

1. Problem Description

The ratio of the interior angle to the exterior angle of a regular polygon is $5:2$. We need to find the number of sides of the polygon.

2. Solution Steps

Let the interior angle be $5x$ and the exterior angle be $2x$.

We know that the sum of an interior angle and its corresponding exterior angle is $180^{\circ}$. Therefore,

$5x + 2x = 180^{\circ}$

$7x = 180^{\circ}$

$x = \frac{180}{7}$

The exterior angle is $2x = 2 \times \frac{180}{7} = \frac{360}{7}$.

The sum of the exterior angles of a polygon is $360^{\circ}$.

Let $n$ be the number of sides of the polygon. Since the polygon is regular, all exterior angles are equal.

Thus, $n \times \frac{360}{7} = 360$

$n = \frac{360}{\frac{360}{7}}$

$n = 360 \times \frac{7}{360}$

$n = 7$

The number of sides of the polygon is

7.

3. Final Answer

The number of sides of the polygon is 7.