A hierarchical finite element for the vibration of membranes is presented. The element has the ability to be joined to neighbouring elements and the numbers of hierarchical terms are allowed to vary in both directions of the element co-ordinate axes. The element transverse displacement is described by four linear shape functions plus a variable number of hierarchical functions which are forms of Legendre orthogonal polynomials. The four nodal displacements and the amplitudes of the hierarchical functions on the edges and in the interior of the element are used as generalized co-ordinates. Inter-element compatibility is achieved by matching the generalized co-ordinates at the nodes and edges shared by elements. Results are obtained for simply supported square and L-shaped membranes. Comparisons are made with exact solutions for the square membrane and with highly accurate approximate and linear finite element solutions for the L-shaped membrane. The results for the square membrane confirm that the solutions always converge from above to the exact values as the numbers of hierarchical terms are increased and highly accurate answers are obtained despite the use of a very few hierarchical terms. The results of the L-shaped membrane show that the hierarchical finite element solutions are largely more accurate than the linear finite element solutions despite the use of fewer system degrees of freedom.