2014 AMC 10A Problem 12

Note that we can split the hexagon up into $6$ equilateral triangles each with side length $6.$ Recall that the area of an equilateral triangle with side length $s$ is $$\dfrac{s^2 \sqrt{3}}{4}.$$ This means that the area of the hexagon is $$6 \cdot \dfrac{6^2 \sqrt{3}}{4} = 54\sqrt{3}.$$ Since each interior angle of a regular hexagon is $120^{\circ},$ the six sectors form $2$ full circles. This means that the area of all the sectors is $$2 \cdot 3^2 \pi = 18\pi.$$ The area of the shaded region is then $$54\sqrt{3} - 18\pi.$$ Thus, C is the correct answer.