HMMT February

Geometry Difficulty 5.0 AIME, harder Prove it United States

Problem:

Let $A B C D E F$ be a regular hexagon with side length $2$ . A circle with radius $3$ and center at $A$ is drawn. Find the area inside quadrilateral $B C D E$ but outside the circle.

Solution

Solution:

Rotate the region $6$ times about $A$ to form a bigger hexagon with a circular hole. The larger hexagon has side length $4$ and area $24 \sqrt{3}$ , so the area of the region is $\frac{1}{6}(24 \sqrt{3}-9 \pi)=4 \sqrt{3}-\frac{3}{2} \pi$ .

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