In the diagram, $P Q R S$ is a square with side length 2 . Each of $P, Q, R$, and $S$ is the centre of a circle with radius 1 . What is the area of the shaded region?
(A) $16-\pi^{2}$
(B) $16-4 \pi$
(C) $4-4 \pi$
(D) $4-4 \pi^{2}$
(E) $4-\pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-064.jpg?height=403&width=404&top_left_y=926&top_left_x=1294)
The area of the shaded region is equal to the area of square $P Q R S$ minus the combined areas of the four unshaded regions inside the square.

Since square $P Q R S$ has side length 2 , its area is $2^{2}=4$.

Since $P Q R S$ is a square, then the angle at each of $P, Q, R$, and $S$ is $90^{\circ}$.

Since each of $P, Q, R$, and $S$ is the centre of a circle with radius 1, then each of the four unshaded regions inside the square is a quarter circle of radius 1 . (A central angle of $90^{\circ}$ gives a quarter of a circle.)

Thus, the combined areas of the four unshaded regions inside the square equals four quarters of a circle of radius 1 , or the area of a whole circle of radius 1 .

This area equals $\pi(1)^{2}=\pi$.

Therefore, the shaded region equals $4-\pi$.

ANSWER: (E)