, 2021

Geometry Difficulty 4.1 AIME Find the answer United States

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?

Pick one

Solution

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
$2 \cdot 6 \cdot \frac{1^2 \cdot \sqrt{3}}{4} - \pi \cdot 1^2 = 3\sqrt{3} - \pi.$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.