Question

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

$\dfrac{\pi}{16}$

$\dfrac{\pi}{8}$

$\dfrac{3\pi}{16}$

$\dfrac{\pi}{4}$

$\dfrac{\pi}{2}$

Solution:

Let the smaller circle have radius $r,$ so its area is $\pi r^2.$

The smaller square, which circumscribes this circle, has side $2r,$ and its diagonal $2\sqrt2\,r$ is the diameter of the larger circle. So the larger circle has radius $\sqrt2\,r.$

The larger square circumscribes the larger circle, so it has side $2\sqrt2\,r$ and area $8r^2.$

The desired ratio is $\dfrac{\pi r^2}{8r^2} = \dfrac{\pi}{8}.$

Thus, B is the correct answer.

2005 AMC 10B Problem 7

Solution:

Problem 7 in Other Years