Answer
$439.34$
Work Step by Step
The area of a triangle with sides of lengths $a$ and $b$ and with included angle $\theta$ is
$A_{t}=\displaystyle \frac{1}{2} ab \sin\theta$
The area $A$ of a sector with central angle of $\theta$ radians is
$ A_{s}=\displaystyle \frac{1}{2}r^{2}\theta$.
The area of a circle with radius r is $A_{c}=\pi r^{2}$
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(shaded area)= (circle) - (sector) + (triangle)
$A=A_{c}-A_{s}+A_{t}$
$=12^{2}\displaystyle \cdot\pi-\frac{1}{2}\cdot 12^{2}\cdot\frac{\pi}{3}+\frac{1}{2}(12)(12)\sin\frac{\pi}{3}$
$=\displaystyle \frac{144}{2}(2\pi-\frac{\pi}{3}+\frac{\sqrt{3}}{2})$
$\approx$439.344947503$\approx 439.34$
