A 4-noded membrane element with drilling degrees of freedom is developed. A functional for the linear problem, in which the drilling rotations are introduced as independent variables, is obtained from the functional for the corresponding nonlinear problem, initially derived by Atluri. A careful examination of the Euler equations shows that the Allman-type element fails to pass a one-element patch test with a minimum number of constraints. A 4-noded element, which passes the one-element patch test without constraining the drilling rotations, is proposed here with the use of the separate kinematics variables of displacement and rotation. Detailed numerical studies show the excellent performance of the element.