What the answer is and how to do it

The line extending from the circle's center to the line AB, splits it into two. Let's name the middle point of line AB, "M." And let's name the center of the circle "O."

So, AM = 3*sqrt3, and 
BM = 3*sqrt3.

It's important to note that the line OM, bisects the angle AOB.

The final piece of info we need is the length OM, but fortunately, the right triangles are all 30-60-90 right triangles, so OM is 3.

And also, AOB is 120 degrees, as shown in the diagram.

The question also tells us that the shaded area is 1/3 of the total circle, since 120/360 = 1/3. 

The area of a circle is r^2*, and the radius of this circle is the hypotenuse of the right triangles. AO = 6, so the area of the circle is