We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities of condensers (G, E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of E. Our computational experiments demonstrate, for instance, sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed