Here's how I'd do it:
Call $R$ the radius of the big circle and $r$ that of the small one. Now, observe that the surface of the crescent is just the difference between the surface of the big circle and the small one or
$\pi R^2 - \pi r^2 = \pi (R-r)(R+r) \; .$
Note how I've expressed this surface as a product of two quantities which I am now going to determine from the other data in the drawing. First the difference between the double radii is clearly 9cm:
$2R-2r=9$
We're halfway. Then the distance between the center of the small circle and point E is obviously $r$, but can alternatively be expressed with Pythagoras as
$r^2 = (R-5)^2+(r-R+9)^2 \; .$
Reordering and using what we already know about $R-r$:
$r^2 - (R-5)^2= \left(\frac{9}{2}\right)^2 \; .$
Again, using the factorizing trick
$(r - R+5)(r+R-5)= \left(\frac{9}{2}\right)^2 \; .$
Thus,
$r+R= 5+2\left(\frac{9}{2}\right)^2 \; .$
Combining everything, we get that the surface of the crescent is
$\pi \frac{9}{2}\left(5+2\left(\frac{9}{2}\right)^2\right)$