Tensegrity structure is a prestressed self-equilibrated system consisting of compressed struts and tensioned cables. Planar truss is one of the most common forms of engineering structures with a wide application. This paper proposes a new class of planar tensegrity trusses whose prototype is unintentionally found by a topology optimization algorithm. The prototype truss is composed of two basic units which are combined alternately. In this paper, the state of prestress of the truss is analytically determined from the equilibrium conditions. The stiffness and strength of the structure are analytically formulated by using the principle of virtual work. Parametric analysis of geometrical parameters on the stiffness and strength of the structure is carried out. By changing the shape of the basic units, the uniform planar tensegrity truss is extended to a fish-bellied truss. Parametric analysis is also carried out for the fish-bellied truss. An example of extending the planar tensegrity truss into a three-dimensional latticed tensegrity truss is also given.