Question

Find the measure of arc DF in degrees, given that the shaded region has an area of 45.27 square centimeters. If necessary, round your answer to two decimal places.

Answer 1

$\bf \textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360}\qquad \begin{cases} r=radius\\ \theta =angle\ in\\ \qquad degrees\\ ----------\\ r=6\\ A=45.27 \end{cases}\implies 45.27=\cfrac{\theta\pi \cdot 6^2}{360} \\\\\\ 45.27\cdot 360=\theta \pi 6^2\implies \cfrac{45.27\cdot 360}{36\pi }=\theta \\\\\\ 144.09888547540\approx\theta \implies 144.10^o\approx\theta =\widehat{DF} \\\\\\ \textit{now, that gives you the central angle, however}\qquad \theta =\widehat{DF}$

bearing in mind that, arcs get their angle measurement from the central angle they're in.  Thus, the measure of that central angle, is the same measure in degrees of that arcDF.