The ratio of the interior to the exterior angles of a regular polygon is 5:2. Find the no. of sides of the polygon

The problem states that the ratio of the interior angle to the exterior angle is $5:2$.

$$\frac{I}{E} = \frac{5}{2}$$

This means $I = \frac{5}{2}E$.

$$I + E = 180^\circ$$

Substitute the expression for $I$ from Step 1 into this equation:

$$\frac{5}{2}E + E = 180^\circ$$ $$\frac{5}{2}E + \frac{2}{2}E = 180^\circ$$ $$\frac{7}{2}E = 180^\circ$$

$$E = \frac{180^\circ \times 2}{7}$$ $$E = \frac{360^\circ}{7}$$

$$E = \frac{360^\circ}{n}$$

Now, equate the value of $E$ found in Step 3 with this formula:

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

$$n = 7$$

The number of sides of the polygon is 7.

The number of sides of the polygon is $\text{7}$.