In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli are conformally equivalent iff the ratio of the outer radius and the inner radius is the same for the two. Thus each is conformally equivalent to a unique "standard" annulus $r < |z| < 1$ with $ 0 < r < 1$. Now my question is the following:

Is there any way to see what the radius of a "standard" annulus conformally equivalent to a doubly connected region is, just by looking at the region? That is without constructing an explicit map.