2005 AMC 8 Problem 25

Question

A square with side length

2

and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

\dfrac{2}{\sqrt{\pi}}

\dfrac{1+\sqrt{2}}{2}

\dfrac{3}{2}

\sqrt{3}

\sqrt{\pi}

Answer: A

Concepts:circle area square (geometry)

Difficulty rating: 1560

Small hint:

The two outside-area totals are equal.

Big hint:

That makes the square and circle have equal total area.

Solution:

Let

S

be the common area inside both the square and circle. The problem says the circle-only area equals the square-only area.

Adding

S

to both equal areas shows that the total area of the circle equals the total area of the square.

The square area is

2^2=4

, so

\pi r^2=4

. Hence

r=\dfrac{2}{\sqrt{\pi}}

.

Thus, A is the correct answer.