The free vibration analysis of membranes is of significant importance in engineering structures, especially in the design and optimization of membrane structures. This paper presents a new type of triangular element, aiming to improve the computational accuracy in free vibration analysis of membranes. Traditional 3-node triangular elements in membrane vibration analysis typically rely on polynomial shape functions, but this method often suffers from insufficient accuracy in complex vibration modes and high-order frequencies. To address this issue, this paper constructs a 10-node triangular element with the shape function incorporating trigonometric functions. The proposed 10-node triangular element consists of three corner nodes, two points of trisection for every edge, and the centroid node with its shape functions derived using the area coordinate method. The stiffness matrix and mass matrix are derived, and the frequencies and modes for free vibration of the membrane are computed, thereby the dynamic characteristics can be studied. To evaluate the effectiveness of this element, several typical examples are chosen, including the free vibration analysis of rectangular membrane and triangular membrane. By comparing with theoretical solutions and the 3-node element calculations in Ansys, the obtained results show that the 10-node triangular element can approximate the theoretical solutions with few computational elements. And the precision of the presented 10-node element is similar with that of the standard 10-node triangular element. The high precision of the proposed element is demonstrated in analysis of free vibration of membrane, which has the potential of further research and promotion.