Question

Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?

$3 - 2\sqrt{2}$

$2 - \sqrt{2}$

$4(3 - 2\sqrt{2})$

$\dfrac{1}{2}(3 - \sqrt{2})$

$2\sqrt{2} - 2$

Solution:

Let each small circle have radius $1.$ Their centers form a square of side $2,$ whose diagonal is $2\sqrt2.$

The large circle's diameter is $2 + 2\sqrt2,$ so its radius is $1 + \sqrt2.$

The desired ratio is $\dfrac{4 \cdot \pi (1)^2}{\pi (1 + \sqrt2)^2} = \dfrac{4}{3 + 2\sqrt2} = 4(3 - 2\sqrt2).$

Thus, the correct answer is C.

2009 AMC 10A Problem 21

Solution:

Problem 21 in Other Years