2011 AMC 8 Problem 25

Question

A circle with radius

1

is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle’s shaded area to the shaded area between the two squares?

\dfrac{1}{2}

1

\dfrac{3}{2}

2

\dfrac{5}{2}

Answer: A

Concepts:circle area square (geometry)Pythagorean Theorem

Difficulty rating: 1660

Small hint:

The desired ratio is

\frac{\pi-2}{2}

, then compare to the choices.

Big hint:

Find the circle area and both square areas.

Solution:

The circle’s shaded area is equal to the area of the circle minus the area of the smaller square. The side length of the inner square can be calculated using the Pythagorean Theorem to get

\sqrt{1^2 + 1^2} = \sqrt{2}.

Therefore, the area of the inner square is

\sqrt{2}^2 = 2.

The circle’s shaded area is then

1^2\pi - 2 = \pi - 2.

The area of the outside square is

2^2 = 4,

so the shaded area between the two squares is

4 - 2 = 2.

The desired fraction is

\dfrac{\pi - 2}{2} \approx \dfrac{3.14 - 2}{2} \approx \dfrac{1}{2}.

Thus, A is the correct answer.