Let \(R(D,a)\) denote the conformal radius of the simply connected domain \(D\) with respect to the point \(a\in D\). Let \(D(R_0)\) denote the set of all simply connected domains \(D\) in the complex plane with \(0,1\in D\) and for which \(R(D,0)\) has the prescribed value \(R_0\). The author poses and solves the problem of finding, in the set \(D(R_0)\), the domain \(D\) for which the conformal radius \(R(D,1)\) has the greatest value. The author then poses and solves the analogous problem for doubly connected domains. (For a doubly connected domain \(D\) with \(a\in D\), one needs to work with simply connected domains \(\widetilde D\subset D\), \(a\in \widetilde D.)\) Quadratic differentials and extremal decompositions play an important part in the proofs.