In the diagram, $P Q$ and $R S$ are diameters of a circle with radius 4 . If $P Q$ and $R S$ are perpendicular, what is the area of the shaded region?
(A) $16+4 \pi$
(B) $8+8 \pi$
(C) $8+4 \pi$
(D) $16+16 \pi$
(E) $16+8 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-088.jpg?height=344&width=357&top_left_y=1774&top_left_x=1277)
Diameters $P Q$ and $R S$ cross at the centre of the circle, which we call $O$.

The area of the shaded region is the sum of the areas of $\triangle P O S$ and $\triangle R O Q$ plus the sum of the areas of sectors $P O R$ and $S O Q$.

Each of $\triangle P O S$ and $\triangle R O Q$ is right-angled and has its two perpendicular sides of length 4 (the radius of the circle).

Therefore, the area of each of these triangles is $\frac{1}{2}(4)(4)=8$.

Each of sector $P O R$ and sector $S O Q$ has area $\frac{1}{4}$ of the total area of the circle, as each has
central angle $90^{\circ}$ (that is, $\angle P O R=\angle S O Q=90^{\circ}$ ) and $90^{\circ}$ is one-quarter of the total central angle.

Therefore, each sector has area $\frac{1}{4}\left(\pi\left(4^{2}\right)\right)=\frac{1}{4}(16 \pi)=4 \pi$.

Thus, the total shaded area is $2(8)+2(4 \pi)=16+8 \pi$.

ANSWER: (E)