2021 Fall AMC 10B Problems/Problem 11

Problem

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?

Solution

Let the hexagon described be of area $H$ and let the circle's area be $C$ . Let the area we want to aim for be $A$ . Thus, we have that $C-H=H-A$ , or $A=2H-C$ . By some formulas, $C=\pi{r}^2=\pi$ and $H=6\cdot\frac12\cdot1\cdot(\frac12\cdot\sqrt3)=\frac{3\sqrt3}2$ . Thus, $A=3\sqrt3-\pi$ or $\boxed{(\textbf{B})}$ .

~Hefei417, or 陆畅 Sunny from China

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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2021 Fall AMC 10B Problems/Problem 11

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