Determine area of circle by finding its diameter or radius first. How do we determine the length EB, FC, AD? A hexagon has 6 sides, so the sum of the angles is (6-2) x 180 = 720. Therefore, each angle of the hexagon is 120* degrees. Because all sides of the hexagon are equal, each triangle must be an isoscoles triangle (30 + 30 + 120* = 180). Split each triangle by bisecting the 120 degree angle to form 2x 30,60,90 triangles. Because these shapes can be any size in relative proportions, invent a number for one of the sides and compute from there. Let’s choose “4″ for AC. Therfore, “2″ is the height of one of the 30,60,90 triangles. The base is therefore 2/?3 and its hypotenous 4/?3. “2/?3″ is important for determining the diameter.

Now, what is the height of the big center triangle? The triangle must be an equalateral triangle if all its sides are the same (60,60,60). We must split this triangle the same as before to determine its height. It becomes another 30,60,90 triangle. If AC = 4, then AG & GC = 2, GE = 2?3, and obviously AE=4.

Now we have all the information necessary to determine the diameter of the circle: the base of the small triangle plus the height of the big triangle (2/?3 + 2?3 = 2/?3 + 6/?3 = 8/?3). The radius is therefore 4/?3, the area (4/?3)^2 = 16/3 x ? = 16?/3.

Next step is to determine the percentage of the area that is shaded by finding total area of the isocoles triangles and making it the numerator (Small Triangle Area / [16?/3]). The area of the small triangles is easy to find. Use our old information regarding the 30,60,90 small triangles and compute 0.5BH = 0.5(2/?3)(2) = 2/?3; therefore, the area of each isocoles triangles is 2 x 2/?3 = 4/?3 and the sum area of all small triangles is 12/?3.

Now we can solve. (12/?3) / (16?/3) = 36 / 16??3 = 16 x (~3.1 x ~1.7 = ~5.3) = ~91… So, 36 / 90 = (36 x 10/9) / 100 = 360/9 /100 = 40/100 = 40%. Answer choice D is closest.