Due to the crucial role of surface roughness, it has been recently proposed to design optimal and extract geometrical microstructures for practical fabrications of superhydrophobic surfaces. In this work, a paraboloid microtexture is employed as a typical example to theoretically establish a relationship between surface geometry and superhydrophobic behavior for a final optimal design. In particular, based on a thermodynamic approach, the effects of all the geometrical parameters for such a paraboloid microtexture on free energy (FE) and free energy barrier (FEB) as well as equilibrium contact angle (ECA) and contact angle hysteresis (CAH) of a superhydrophobic surface have been systematically investigated in detail. It is interestingly noted that the droplet position for metastable state is closely related to the intrinsic CA of the surface. Furthermore, the paraboloid base steepness plays a significant important role in ECA and CAH, and a critical steepness is necessary for the transition from noncomposite to composite states, which can be judged using a proposed criterion. Moreover, the superhydrophobicity depends strongly the surface geometrical dimension for noncomposite state, while it is not sensitive for composite state. Additionally, both vibrational energy and geometrical dimension affect the transition from noncomposite to composite wetting states, and a comprehensive criterion for such transition can be obtained. Finally, using such criterion, it is revealed that the paraboloidal protrusion is the most optimal geometry among the three typical microtextures for ideal superhydrophobicity.