Question

A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?

$\frac{5\sqrt{3}}{2} - \pi$

$3\sqrt{3}-\pi$

$4\sqrt{3}-\frac{3\pi}{2}$

$\pi - \frac{\sqrt{3}}{2}$

$\frac{\pi + \sqrt{3}}{2}$

Solution:

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 .

2021 AMC 10B Fall Problem 11

Solution:

Problem 11 in Other Years