Two similar regular hexagons have a common center. if each side of the big hexagon is twice the side of the small one and the area of the small hexagon is 3 sq. in, what is the area of the big hexagon?

To determine any area of a shape we need to know the formula for the area of that particular shape which is for a regular hexagon is 

Area of hexagon = (3 √3) a^2 / 2
 where a is the length of the side of the hexagon
So, first, we determine the length of the side of the smaller hexagon from the given area.

3 in^2 = (3 √3) a^2 / 2
6 in^2 = (3 √3) a^2 
a^2 = 2 √3 / 3
a = 1.0746 in

The length of the side of the bigger hexagon is twice that of the smaller so,
A = 2a = 2 (1.0746 in ) = 2.1491 in

Area of the bigger hexagon =  (3 √3) a^2 / 2
                                            = (3 √3) 2.1491^2 / 2                                            = 12 in^2