geometry

Let the measure of an exterior angle of the polygon be $x$. Then $x + 10x = 180^\circ$, so $x = \left(\frac{180}{11}\right)^\circ$. The measure of an exterior angle of a regular polygon with $\ell$ sides is equal to $\left(\frac{360}{\ell}\right)^\circ$. Thus, $\left(\frac{360}{\ell}\right)^\circ = \left(\frac{180}{11}\right)^\circ$, so the polygon has $\boxed{22}$ sides.

Sum of interior anges =   ( n-2) * 180 

     now divide by n to find the measure of each interior angle    (n-2)* 180/n

        the exterior angles are supplementary   ( and exterior is 10 times bigger) 

                                                                            10*  ( 180 - (n-2)*180/n)   =   (n-2)*180/n

                                                                             1800 - 10(n-2)*180/n = (n-2)*180/n

                                                                             1800 = 11 (n-2)*180/n

                                                                             1800n = (11n-22) * 180 

                                                                              -180n = -3960 

                                                                                n = 22 sides