Deployable mechanisms play an increasingly crucial role in aerospace, yet their design still faces challenges: 1) The folded height increases while its deployed height decreases, which is not conducive to improving the folding performance; 2) Compared with revolute joints, prismatic joints have relatively weak reliability in complex space environments. To address the above challenges, novel constant-height deployable mechanisms with only revolute joints are proposed in this paper. The deployable mechanisms exhibit one degree of freedom, enabling them to attain modular expansion. Subsequently, the mobility properties of these deployable mechanisms are substantiated through screw theory. Assembly strategies for constant-height deployable mechanisms are proposed, and deployable ring trusses are further constructed to provide support structures for satellite mesh reflector antennas. The kinematics are analyzed to investigate the deployable characteristics of the proposed deployable mechanisms and ring trusses. The prototypes of a deployable mechanism and a ring truss are established to validate the designed deployable mechanisms. This work contributes novel types of deployable mechanisms with potential advantages, aiming to function as valuable references for the design and analysis of deployable trusses.