Two 4-node quadrilateral membrane elements, denoted as AGQ6-I and AGQ6-II, have been developed in this paper. Instead of the traditional isoparametric coordinate, the quadrilateral area coordinates were used to establish the formulations of the new elements. And several generalized conforming conditions were then introduced to determine all unknown parameters. Numerical examples showed that the presented elements exhibit excellent performances in both regular and distorted mesh divisions. They could even yield exact solutions for pure bending problems under distorted meshes and provide lock-free solutions for the MacNeal's test problem of trapezoidal locking. Besides, the weak patch test was conducted to guarantee the convergence of both new elements. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements.