We have computed the shape of an axisymmetric fluid/liquid meniscus in a capillary that is spun about its horizontal axis. The solutions were used to establish the relationship between the fluid/liquid/solid contact angle on the one hand and several experimentally accessible parameters (tube radius, interfacial tension, densities, angular velocity, and certain meniscus dimensions) on the other. This opens the way to a novel method for determining contact angles. Using the results for single menisci, vie have in addition developed a method whereby the contact angle may be obtained from the critical speed at which a finite drop (or "slug") is either released from the tube wall (if the drop phase is the less dense phase) or penetrated axially by the other phase (if the drop phase is the denser phase).