The time required to drain process or storage vessels of their liquid contents can be of crucial importance in many emergency situations. Over the years, many formulas have been developed to compute the time requirements for draining vessels of various geometric configurations. These formulas typically assume gravity drainage through an orifice type of opening at the bottom of the vessel. In this article, analytical expressions are developed to compute the times required for complete drainage of annuli (both horizontal and vertical), as well as horizontal tori, solely under the influence of gravity. Annuli often form the shell side of double-pipe heat exchange devices, while horizontal tori are often employed as distributing rings or manifolds in various mass transfer operations. Large vertical tori are sometimes used as promotional devices for advertising purposes (e.g., huge tires), but are not commonly used to contain fluids (presumably because of structural and other considerations). The drainage time expression for horizontal annuli incorporates elliptic integrals.