Question

The large circle has diameter $AC.$ The two small circles have their centers on $AC$ and just touch at $O,$ the center of the large circle. If each small circle has radius $1,$ what is the value of the ratio of the area of the shaded region to the area of one of the small circles?

between $\dfrac12$ and $1$

$1$

between $1$ and $\dfrac32$

between $\dfrac32$ and $2$

cannot be determined from the information given

Solution:

The large circle has radius $2,$ so its area is $\pi(2)^2 = 4\pi,$ and each small circle has area $\pi.$

By symmetry the shaded region is half the difference of the areas: $\dfrac12(4\pi - 2\pi) = \pi.$ The ratio of this to one small circle's area $\pi$ is $1.$

Thus, the correct answer is B .

1986 AMC 8 Problem 23

Solution:

Problem 23 in Other Years