mensuration is hard...

The length of the circular arc is $38-9-9=20~cm$ . The formula for the length of a circular arc is $c=\dfrac{\theta}{360}(2\pi r)$ where $\theta$ is the central angle and $r$ is the radius. Substituting we get

$20=\dfrac{\theta}{360}(2\pi)(9)$

$\theta \approx 127.32$

The area of a circular sector is given by $A=\dfrac{\theta}{360}\pi r^2$ , substituting, we get

$A=\dfrac{127.32}{360} \times \pi (9^2) =\boxed{90}$