Question

How do you figure this out?

Answer 1

To find the area of the shaded region, one must know the areas of the segment and the circle. We have radius which will give us access to area of circle and we have the angle 60 degrees with top of angle being the center of the circle and that will give us access to the area of the segment. Then we take take the difference and get area of shaded region.

So, first area of circle,

$A_c=\pi r^2=\pi5^2=25\pi$

Then the area of the segment,

$A_s=\frac{1}{2}r^2(\theta-\cos(\theta))=\frac{1}{2}5^2(\pi/3-\cos(\pi/3)=12.5(\pi/3-\frac{1}{2})=12.5\cdot\frac{2\pi-3}{6}=6.84$.

Now subtract,

$A=A_c-A_s=25\pi-6.84=71.7$

Hope this helps.