In the diagram, the circle has centre $O$. The shaded sector $A O B$ has sector angle $90^{\circ}$, and $A B$ has arc length $2 \pi$ units. The area of sector $A O B$ is
(A) $4 \pi$
(D) $24 \pi$
(B) $16 \pi$
(C) $6 \pi$
(E) $8 \pi$

![](https://cdn.mathpix.com/cropped/2024_04_20_e85ef69091c57cb633f3g-130.jpg?height=349&width=376&top_left_y=1590&top_left_x=1357)
Let $r$ be the radius of the circle.

Since $\angle A O B=90^{\circ}$, then this sector is one quarter of the whole circle, so the circumference of the circle is $4 \times 2 \pi=8 \pi$.

So $2 \pi r=8 \pi$, or $r=4$.

Thus the area of sector $A O B$ is $\frac{1}{4} \pi r^{2}=\frac{1}{4} \pi(16)=4 \pi$.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-234.jpg?height=412&width=458&top_left_y=1317&top_left_x=1300)

ANSWER: (A)