Question

Find the area of the shaded segment of the circle.

Answer 1

since we know the arc made by the central angle is 120°, thus the central angle is obviously 120°.

$\bf \textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left( \cfrac{\pi \theta }{180}- sin(\theta ) \right)~~ \begin{cases} \theta =central~angle\\ \qquad in~degrees\\ r=radius\\[-0.5em] \hrulefill\\ \theta =120\\ r=6 \end{cases} \\\\\\ A=\cfrac{6^2}{2}\left( \cfrac{\pi (120)}{180}-sin(120^o) \right)\implies A=\cfrac{36}{2}\left(\cfrac{2\pi }{3}-\cfrac{\sqrt{3}}{2} \right) \\\\\\ A=18\left(\cfrac{2\pi }{3}-\cfrac{\sqrt{3}}{2} \right)\implies A\approx 22.11$