Square $A B C D$ with side length 2 is inscribed in a circle, as shown. Using each side of the square as a diameter, semicircular arcs are drawn. The area of the shaded region outside the circle and inside the semi-circles is
(A) $\pi$
(B) 4
(C) $2 \pi-2$
(D) $\pi+1$
(E) $2 \pi-4$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-135.jpg?height=347&width=355&top_left_y=1144&top_left_x=1357)

## Part C: Each correct answer is worth 8.
Square $A B C D$ with side length 2 is inscribed in a circle, as shown. Using each side of the square as a diameter, semicircular arcs are drawn. The area of the shaded region outside the circle and inside the semi-circles is
(A) $\pi$
(B) 4
(C) $2 \pi-2$
(D) $\pi+1$
(E) $2 \pi-4$

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-246.jpg?height=382&width=390&top_left_y=617&top_left_x=1431)

## Solution

The side length of the square is 2 and thus the diameter of the circle has length $2 \sqrt{2}$ which is also the length of the diagonal $A C$ (or $B D$ ). The area of the circle is thus $\pi(\sqrt{2})^{2}=2 \pi$. Since the side length of the square is 2 , it will have an area of 4 . From this, we calculate the area of the circle outside the square to be $2 \pi-4$. To calculate the shaded area, we first calculate the area of each semi-circle. Each of the semi circles has a radius of 1 meaning that each semi-circle will have an area of $\frac{1}{2}\left[\pi(1)^{2}\right]=\frac{1}{2} \pi$. In total, the four semi-circles have an area of $2 \pi$. Thus the shaded area has an area of $2 \pi-(2 \pi-4)=4$.

ANSWER: (B)

## Part C