This paper introduces a new planar flexible hinge of fractal configuration to be incorporated in out-of-plane-motion compliant stages that cover a wide stiffness range. The large-displacement fractal hinge consists of a series of scaled-down, concentric, circular-axis flexible segments that are connected in a folded manner by radial rigid links. In-plane and out-of-plane compliance matrices are derived for fractal hinges with variable planar geometry features. The flexible hinges are assembled in a radial architecture to form compliant stages that can be utilized for piston-type, out-of-plane sensing or actuation. The analytical model calculates the stage active, out-of-plane stiffness, as well as its parasitic, in-plane stiffness. The stage analytical stiffness is confirmed by the experimental testing of a prototype and by finite element simulation. Furthermore, analytical-model simulation is performed to evaluate the variations in the active and parasitic stiffnesses with key geometric parameters, which also enables optimization.