If we find the area of the big circle, and subtract the area of the triangle, we'll get the total area of the three equal sections outside of the triangle and inside the big circle. We only want the area of one of those sections, so if we do that subtraction, we'll need to divide by 3. So we want to find:
(Area of big circle - Area of triangle)/3
Now if you look at the small triangle at the top right of the diagram, the one with the dotted lines as two of its edges, notice that the line of length m in that triangle must meet the big equilateral triangle at 90 degrees, because the edge of the equilateral is tangent to the circle (and tangent lines are always perpendicular to radii at the tangent point). So that dotted-line triangle is a right triangle, and since we're bisecting the 60 degree angle at the top of the diagram, it must be a 30-60-90 triangle, and its sides are in a 1 to √3 to 2 ratio. Its shortest side is m, so its longest side, the vertical dotted line, is 2m. But that's the radius of the big circle, so the area of the big circle is π (2m)^2 = 4πm^2
Notice also that the third side of that small triangle, the downwards sloping side on the right, has length √3m. But that's half the length of a side of the equilateral triangle, so each side of the equilateral has length 2√3m. We need the area of that triangle, so we need to find a height (unless you know an equilateral triangle formula). If we take the horizontal side at the bottom as our base, then the vertical height is just made up of one radius of the small circle, of length m, plus the radius of the big circle which we found above was 2m. So the height of the triangle is 3m. Since our base is just the length of a side in the equilateral triangle, our base is 2√3m, and the area of the equilateral triangle is bh/2 = (2√3m)(3m)/2 = 3√3m^2.
Finally,
(Area of big circle - Area of triangle)/3 = (4πm^2 - 3√3m^2)/3 = (4π - 3√3)m^2/3
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com