Question

the measure of central angle ycz is 80 degrees.
what is the sum of the areas of the two shaded sectors?
$\bigcirc$ $18pi$ units$^{2}$
$\bigcirc$ $36pi$ units$^{2}$
$\bigcirc$ $45pi$ units$^{2}$
$\bigcirc$ $81pi$ units$^{2}$

Explanation:

Step1: Find total shaded angle

First, note that $\angle XCW$ is vertical to $\angle YCZ$, so $\angle XCW = 80^\circ$. Total shaded angle: $80^\circ + 80^\circ = 160^\circ$

Step2: Find circle area

The radius $r=9$. Circle area formula: $A_{circle} = \pi r^2$
$A_{circle} = \pi (9)^2 = 81\pi$

Step3: Calculate shaded area sum

Shaded area is $\frac{\text{total shaded angle}}{360^\circ}$ of the circle.
$\text{Shaded Area} = 81\pi \times \frac{160}{360}$
Simplify $\frac{160}{360} = \frac{4}{9}$, so $81\pi \times \frac{4}{9} = 36\pi$

Answer:

$36\pi$ units$^2$ (Option B)