Is this enough?

Let $O$ be the shared center of the concentric circles and $P$ be the point where $ST$ is tangent to the smaller circle. Also, let $R$ and $r$ be the radii of the larger and smaller circles, respectively.

Now triangle $OPS$ is a right-angled triangle with shorter sides length $r$ and $\frac{1}{2} ST = 18$ and hypotenuse $R$ . So by Pythagoras we have that $R^{2} - r^{2} = 18^{2} = 324$ .

But the area of the annulus is just $\pi*R^{2} - pi*r^{2} = \pi*(R^{2} - r^{2}) = 324\pi$ , and so $a = \boxed{324}$ .

A trivial solution is to reduce the inner circle to a point (i.e. a circle with a radius of zero). That makes the radius of the outer circle 18 and the answer 324. This solution doesn't demonstrate that we have enough data to get the answer (a major flaw). However, if we assume that we do have enough data, it does give the correct answer.

if r < R are the two radius of the circles, and:

|ST| = 2L = 36; L = 18;

Area Shaded = Ash = π(R^2 - r^2);

by pythagoran theorem:

R^2 - r^2 = L^2 = 18^2 = 324

Ash = 324π = 1017.876019763...

It's somewhat well known that the formula for an incircle is 1/12 pi x^2 , and the area of a circumcircle is 1/3 pi x^2. What remains is some simple plugging in, and one gets the result of 1/3 pi (36)^2 - 1/12 pi (36)^2 = a pi

which comes out to 324