Question 250513
i = measure of the interior angle of regular polygon.
e = measure of the exterior angle of regular polygon.


i = 6*e + 12


s = number of sides in regular polygon.


exterior angle = 360 divided by number of sides = 360/s


interior angle = (s-2)*180 / s


example:


exterior angle of regular triangle = 360 / 3 = 120


interior angle of regular triangle = 1*180/3 = 60


the interior angle of a polygon and its associated exterior angle are supplementary to each other.  the sum of their angles is 180.


we get i + e = 180


we also get:


i = 6*e + 12


substitute in supplementary equation to get:


6*e + 12  + e = 180


combine like terms to get:


7*e + 12 = 180


subtract 12 from both sides of equation to get:


7*e = 180 - 12 = 168


divide both sides of equation by 7 to get:


e = 168/7 = 24.


i = 6*24 + 12 = 156


156 + 24 = 180 so the angles are supplementary as they should be.


we have:


i = 156
e = 24


formula for exterior angle of a regular polygon is:


e = 360/s


this becomes


24 = 360/s


solve for s to get:


s = 15


number of sides in the regular polygon appears to be 15.


substitute in equation for interior angle of polygon to get:


156 = (s-2)*180/s


solve for s.


mulltiply both sides of the equation by s to get:


156*s = (s-2)*180


simplify by removing parentheses to get:


156*s = 180*s - 360


solve for s to get:


24*s = 360


s = 360/24 = 15


everything checks out so the answer is:


number of sides of the regular polygon is 15.