Question

A sector of a circle has central angle 30 degrees and arc length 4 cm. Find the area of the sector in terms of pi and rounded to the nearest square cm.

Answer 1

`\textit{Arc's Length}\\\\ s = \cfrac{\theta \pi r}{180} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ s=4\\ \theta =30 \end{cases}\implies 4=\cfrac{(30)\pi r}{180}\implies 4=\cfrac{\pi r}{6} \\\\\\ 24=\pi r\implies \cfrac{24}{\pi }=r \\\\[-0.35em] ~\dotfill\\\\`

`\textit{Area of a Sector of a Circle}\\\\ A=\cfrac{\theta \pi r^2}{360} ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ \theta =30\\ r=(24)/(\pi ) \end{cases}\implies A=\cfrac{(30)\pi \left( (24)/(\pi ) \right)^2}{360}\implies A=\cfrac{\pi \cdot (24^2)/(\pi ^2)}{12} \\\\\\ A=\cfrac{~~ ( 24^2 )/( \pi ) ~~}{12}\implies A=\cfrac{24^2}{12\pi }\implies A=\cfrac{48}{\pi }\implies A\approx 15~cm^2`