The formula for the area of a regular dodecagon is $3r^2$. The answer is $\boxed{\textbf{(A)}}$.
(If you don't know this formula, it's pretty easy to figure out that the area of a square inscribed in a circle is $2r^2$, and all the choices except $3r^2$ are less than $2r^2$. Remember, the more sides a regular polygon has, the closer its area gets to $\pi r^2$.)