The ratio of the interior angle to the exterior angle of a regular polygon is given as $5:2$. 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 given as $5:2$. 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 the interior and exterior angles at a vertex of a polygon is $180^{\circ}$. Therefore, we can write:

$5x + 2x = 180$

$7x = 180$

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

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

The sum of exterior angles of any polygon is $360^{\circ}$. If the polygon has $n$ sides, then each exterior angle in a regular polygon is $\frac{360}{n}$.

So, we have:

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

$n = 7$

Alternatively, the interior angle is $5x = 5(\frac{180}{7}) = \frac{900}{7}$.

The formula for the interior angle of a regular n-sided polygon is $\frac{(n-2) \times 180}{n}$.

Therefore, $\frac{(n-2) \times 180}{n} = \frac{900}{7}$

$(n-2) \times 180 \times 7 = 900 \times n$

$(n-2) \times 1260 = 900n$

$1260n - 2520 = 900n$

$360n = 2520$

$n = \frac{2520}{360}$

$n = 7$

3. Final Answer

The number of sides of the polygon is 7.