Question

Angle AOC has a measure of 5ππ6 radians.The length of arc AB is 2π and the radius is 12.What is the area of sector BOC?

Answer 1

`\textit{arc's length}\\\\ s = r\theta ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ r=12\\ s=2\pi \end{cases}\implies 2\pi =12\theta \implies \cfrac{2\pi }{12}=\theta \implies \cfrac{\pi }{6}=\theta \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\measuredangle AOC}{\cfrac{5\pi }{6}}~~ - ~~\stackrel{\measuredangle AOB}{\cfrac{\pi }{6}}\implies \cfrac{4\pi }{6}\implies \stackrel{\measuredangle BOC}{\cfrac{2\pi }{3}} \\\\[-0.35em] ~\dotfill`

`\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta r^2}{2} ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ r=12\\ \theta =(2\pi )/(3) \end{cases}\implies A=\cfrac{2\pi }{3}\cdot \cfrac{12^2}{2}\implies A=48\pi \implies A\approx 150.80`