Question

A square has sides of length $10,$ and a circle centered at one of its vertices has radius $10.$ What is the area of the union of the regions enclosed by the square and the circle?

$200 + 25\pi$

$100 + 75\pi$

$75 + 100\pi$

$100 + 100\pi$

$100 + 125\pi$

Answer: B

Concepts:circle area sector inclusion-exclusion

Difficulty rating: 1270

Small Hint:

Add the areas of the square and the circle, then subtract the overlap

Big Hint:

The overlap is the quarter of the circle that lies inside the square

Solution:

The square has area $10^2 = 100$ and the circle has area $\pi(10)^2 = 100\pi.$

Since the circle is centered at a vertex of the square, exactly one quarter of the circle, area $25\pi,$ lies inside the square.

The union has area $100 + 100\pi - 25\pi = 100 + 75\pi.$

Thus, the correct answer is B.

2004 AMC 10B Problem 9

Answer: B

Difficulty rating: 1270

Small Hint:

Big Hint:

Solution:

Problem 9 in Other Years