By "see" I will assume you mean in a geometric sense. Then your question falls within a standard topic in geometric complex analysis. First some terminology: A doubly connected domain $R$ on the Riemann sphere is called a *ring domain*, and if you map it onto $r < |z| < s$ as a canonical domain, then $\mathrm{mod}(R) = (2\pi)^{-1}\log(s/r)$ is called the *conformal modulus* or just *modulus* of the ring domain. By the way I have defined it, it is nearly trivially a conformal invariant, but it need not be defined this way. There is a geometric theory, the Ahlfors-Beurling theory of extremal length of curve families, within which the modulus of a ring domain can be defined directly and geometrically, without any preliminary conformal mapping onto some canonical domain. Extremal length can be proved to be a conformal invariant, and then one quickly sees that the two definitions coincide. There is an exposition of the theory of extremal length in *Conformal Invariants* by Ahlfors.

It would be unreasonable to expect to "see" the *exact* value of the modulus of a ring domain. The boundary of a ring domain can be extremely complex geometrically, and every tiny wiggle impacts on the modulus. But extremal length yields inequalities for the modulus from geometric data. I will quote a single, rather striking, result of this kind:

If a ring domain $R$ contains no circle on the Riemann sphere separating its two boundary components, then $\mathrm{mod}(R) \leq 1/4$. The constant $1/4$ is sharp. The result is due to D. A. Herron, X. Y. Liu and D. Minda.

Small modulus means "thin" ring domain; if the modulus is large enough, the ring domain is so "fat" that it has to contain a separating circle.