Let n be the number of sides of the regular polygon\(\left(n\in N;n\ge3\right)\),then the measure of each interior angle is \(\dfrac{180\left(n-2\right)}{n}\) and the measure of each exterior angle is \(180-\dfrac{180\left(n-2\right)}{n}\).We have :

\(\dfrac{180\left(n-2\right)}{n}=8\left(180-\dfrac{180\left(n-2\right)}{n}\right)\)

\(\Leftrightarrow\dfrac{180\left(n-2\right)}{n}-1440+8.\dfrac{180\left(n-2\right)}{n}=0\)

\(\Leftrightarrow9.\dfrac{180\left(n-2\right)}{n}=1440\Leftrightarrow\dfrac{9\left(n-2\right)}{n}=8\Rightarrow9n-18=8n\Leftrightarrow n=18\)

So,the polygon has 18 sides