The shape of a meniscus of one phase between two others is studied in two dimensions using random walk models. An interface with a meniscus is approximated by two random walks forming microscopic droplets of the intruding phase before and after a macroscopic lens. Within this class of models, we establish a Wulff construction and prove the Herring relations between contact angles. We give explicit formulas for the contact angles as functions of temperature, both at low temperatures and near the wetting transition.