2011 AMC 10A Problem 18

Question

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?

3 - \dfrac{\pi}{2}

\dfrac{\pi}{2}

2

\dfrac{3\pi}{4}

1+\dfrac{\pi}{2}

Answer: C

Concepts:circle area sector area decomposition

Difficulty rating: 1790

Small hint:

Use the unit squares formed by the centers and intersection points

Big hint:

Two quarter-circle subtractions plus a semicircle leave a simple area

Solution:

The area of this region is the area of circle

C

minus the area of the overlapping regions with

A

and

B.

From the diagram, we can find the area of half of one of the overlapping regions by finding the area of the sector and subtracting the area of the triangle.

This area is then

\dfrac{1}{4} \pi \cdot 1^2 - \dfrac{1}{2} \cdot 1 \cdot 1 = \dfrac{\pi}{4} - \dfrac{1}{2}.

There are four of these that we must subtract, which leaves us with a final answer of

\pi \cdot 1^2 - 4(\dfrac{\pi}{4} - \dfrac{1}{2}) = 2.

Thus, C is the correct answer.