Fall 2021 AMC 10 B

Note that the reflected arcs do not overlap except at their endpoints. The area of the region can be found by subtracting from the area of the hexagon the difference between the areas of the circle and the hexagon. This is equivalent to twice the area of the hexagon minus the area of the circle. Therefore the requested area is $$2\cdot 6\cdot \frac{1^2\cdot \sqrt{3}}{4}-\pi \cdot 1^2=3\sqrt{3}-\pi .$$