2011 AMC 10A Problems/Problem 18

Problem 18

Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$ . What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?

Solution

Draw a rectangle with vertices at the centers of $A$ and $B$ and the intersection of $A, C$ and $B, C$ . Then, we can compute the shades area as the area of half of $C$ plus the area of the rectangle minus the area of the two sectors created by $A$ and $B$ . This is $\frac{\pi (1)^2}{2}+(2)(1)-2 \cdot \frac{\pi (1)^2}{4}=\boxed{2 \ \mathbf{(C)}}$ .

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2011 AMC 10A Problems/Problem 18

Problem 18

Solution