Suppose this is the right side of the polygon, with the two
green lines being the extensions of one pair of alternate 
sides.  (We only need to extend one pair of alternate sides). 



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The angle between the two green sides is given 
as 120°. Each of the acute angles are exterior 
angles.  Since the polygon is regular, these two 
acute angles, which are exterior angles, have 
equal measure.  Therefore, the triangle with two 
green sides is isosceles.  Each of its base 
angles is 180°-120° or 60°.  So each base angle 
is half of that or 30°.

Each base angle of 30° is an exterior angle of 
the polygon, and the sum of all exterior angles 
of any polygon is 360°.  All the exterior angles 
of this polygon have the same measure.  So if 
the number of sides is n, then each exterior 
angle equals . So we have the 
equation:

            

Multiply both sides by n:

              
              
                    

So the regular polygon has 12 sides.  Here's a smaller picture of
the 12-sided regular polygon:



Edwin