jmc

Question

A circle of radius 1 is internally tangent to two circles of radius 2 at points

A

and

B

, where

A B

is a diameter of the smaller circle. What is the area of the region, shaded in the figure, that is outside the smaller circle and inside each of the two larger circles? Express your answer in terms of

π

and in simplest radical form.

Accepted Answer

The centers of the two larger circles are at

A

and

B

. Let

C

be the center of the smaller circle, and let

D

be one of the points of intersection of the two larger circles.

Then

△ A C D

is a right triangle with

A C = 1

and

A D = 2

, so

C D = 3

,

∠ C A D = 6 0^{\circ}

, and the area of

△ A C D

is

3 /2

. The area of 1/4 of the shaded region, as shown in the figure, is the area of sector

B A D

of the circle centered at

A

, minus the area of

△ A C D

, minus the area of 1/4 of the smaller circle. That area is

\frac{2}{3} π - \frac{3}{2} - \frac{1}{4} π = \frac{5}{12} π - \frac{3}{2},

so the area of the entire shaded region is

4 (\frac{5}{12} π - \frac{3}{2}) = \frac{5}{3} π - 23 .