We have the following equation to find the area of a circle sector:
[tex]A=(\frac{\theta}{2\pi})\cdot\pi\cdot r^2[/tex]
in this case, we have the following information:
[tex]\begin{gathered} A=27\pi \\ r=9 \end{gathered}[/tex]
using these values on the equation and solving for theta, we get:
[tex]\begin{gathered} (\frac{\theta}{2\pi})\cdot\pi\cdot(9)^2=27\pi \\ \Rightarrow\frac{\theta}{2}\cdot81=27\pi \\ \Rightarrow\frac{\theta}{2}=\frac{27\pi}{81}=\frac{\pi}{3} \\ \Rightarrow\theta=\frac{2}{3}\pi \end{gathered}[/tex]
therefore, the angle is 2/3 pi. Converting to decimal, we get:
[tex]\frac{2}{3}\pi\cdot\frac{180}{\pi}=\frac{360}{3}=120\degree[/tex]
therefore, the angle measures 120 degrees
