Well the area which s common to both the hexagons will be minimum if the angle between the corresponding intersecting sides of two hexagons is 120 degrees. Now draw figure and using simple geometry and elementary trigonometry the minimum area can easily be worked out.

 

Similarly, are will be maximum if the sides overlap that is angle between the corresponding sides of two hexagons is 0 degrees.

 

So as we rotate one hexagon keeping other fixed the area will be mimnimum if the earliar overlapping sides form angle 120 degree.