2021 Fall AMC 10B Problem 11

The original circle is made from the regular hexagon plus $6$ equal circular segments. Reflecting each minor arc over its side puts those same $6$ segments inside the hexagon instead. Therefore the average of the circle’s area and the reflected-arc region’s area is the area of the regular hexagon. The hexagon has area $6\cdot\frac{\sqrt3}{4}=\frac{3\sqrt3}{2},$ and the circle has radius $1,$ so its area is $\pi.$ If the desired area is $A,$ then $\frac{A+\pi}{2}=\frac{3\sqrt3}{2},$ so $A=3\sqrt3-\pi.$ Thus, the answer is B .