In the figure below, $O$ is the center of the circle, and the radius of the circle is $8$. Angle $\angle BOA$ is right. What is the area of the filled-in (purple) region?

Explain the steps of your solution. Give your answer in exact form (which may involve pi). In other words, don't round off (but do simplify if you can).

https://latex.artofproblemsolving.com/4/7/6/476213a2b084de24a8a4577fa8ac90f4c6e672ff.png

To find the shaded region, we need to subtract the area of the right triangle (OAB) from the full sector. We can find the area of this sector by looking at the inner angle, which is 90 degrees. Since 90 is 1/4 of 360, the sector OAB is also 1/4 of the full circle's area.

Can we find the area of the full circle? Since we know the radius (8), yes.

The area of a circle is the radius squared times pi. The radius squared is 64, therefore the area of the circle is 64$\pi$.

64 divided by 4 is 16, so the area of sector OAB is 16$\pi$.

The right triangle is fairly easy to find the area of. Both line segments OA and OB are equal because they are both the radius of the circle, so I can just use them easily as the base and height to get:

8 * 8 = 64, 64/2 = 32.

32 is the area of the right triangle, so we can finally find the area of the shaded part by subtracting:

16$\pi$-32 units squared.