Fly the kite ...

It is assumed that $DC$ and $BC$ are tangents to the circular sector. So $\angle ADC= \angle ABC=90^\circ$ . It follows that $\angle BCD=60^\circ$ . Draw line $BD$ . Compute the length of line $BD$ by cosine law. Compute the area of $\triangle DAB$ and $\triangle DCB$ using the formula $\dfrac{1}{2} ab \sin C$ . Take the sum, it is approximately $22.44737847$ . Compute the area of the circular sector using the formula $\dfrac{\theta}{360} \pi r^2$ . It is approximately $13.57168026$ . The difference of the two areas is the area of the shaded region, we have

$A= 22.44737847-13.57168026=\boxed{8.875698203}$