If in the figure above, AB is the diameter of a circle of radius 6, wh

If you connect a diameter to a point on a circle, you always get a right angle at that point. So if we draw the triangle ABC here, it will have a 90 degree angle at C, and will therefore be a 30-60-90 triangle. It has a hypotenuse of length 12 (twice the radius), and since the sides of any 30-60-90 triangle are in a 1 to √3 to 2 ratio, the other two sides must be 6 and 6√3. The two longest of those sides, 6√3 and 12, are edges of the shaded region. So the answer is 12 + 6√3 + the length of the minor arc BC. So the answer is B or D. But the arc BC is clearly (slightly) longer than the straight line BC, which we just found was of length 6. From our remaining two answer choices, we can see that its length must be 2π, and not π, so the right answer must be D.

Of course you can also find the exactly length of that arc in other ways (for example, if O is the center of the triangle, it's easy to see that the triangle OCB will be equilateral, since two sides are radii and we found the third side CB above is also 6; that makes the angle COB a 60 degree angle, and minor arc BC is thus 60/360 = 1/6 of the circumference of the circle) but an estimate saves a bit of time.