In a certain regular polygon, the measure of each interior angle is \(10\) times the measure of each exterior angle. Find the number of sides in this regular polygon.
+26 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.
+37158 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
