The radius of the circle is 11, so the area is
[tex]A=\pi r^2 = 121\pi[/tex]
The central angles of the shaded and non-shaded regions sum up to 360 degrees, so the central angle of the shaded region is
[tex]360-217=143[/tex]
The area of the shaded region is in proportion with the area of the whole circle: if the whole area is given by a sector of 360°, the area of a 143° sector will be given by
[tex]A_{360}\div A_{143} = 360\div 143[/tex]
Since we know that the whole area is [tex]121\pi[/tex], we can solve for the area of the 143° sector:
[tex]121\pi\div A_{143} = 360\div 143 \iff A_{143}=\dfrac{121\pi\cdot 143}{360} \approx 151[/tex]
