The area of a circular sector is proportional to the central angle that it encloses.
Since the non-shaded region encloses an angle of 200°, then the shaded region must enclose an angle of 160°, so that 200+160=360.
Multiply the area of a complete circle by 160/360 to find the area of the shaded sector.
The area of a circle is given by:
[tex]A=\pi r^2[/tex]
Then, the area of a circular sector that encloses an angle k measured in degrees, is:
[tex]A=\pi r^2\cdot\frac{k}{360}[/tex]
In the given diagram, we can see that the radius equals 9yd. Then, the area of the shaded sector is:
[tex]\begin{gathered} A=\pi(9yd)^2\cdot\frac{160}{360} \\ =\pi\cdot81\cdot\frac{4}{9}yd^2 \\ =36\pi yd^2 \end{gathered}[/tex]
Therefore, the area of the shaded sector is:
[tex]36\pi yd^2[/tex]
