CEMC Pascal

The area of the shaded region is equal to the area of square $PQRS$ minus the combined areas of the four unshaded regions inside the square. Since square $PQRS$ has side length 2, its area is $2^{2}=4$. Since $PQRS$ 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. 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 $4-\pi$.