This paper presents the design and development of a polar-structure 3D printing system specifically tailored for clay material. Unlike conventional Cartesian or SCARA-based printers, which move the nozzle to designated locations, the proposed system combines the movement of a rotation table and translation along the X-axis to print layers of the model. A screw-based extruder driven by a worm gear provides precise control over extrusion force, and the kinematics of the polar system are mathematically analyzed to derive a dynamic relationship between angular position and displacement radius. This configuration enables synchronization between construction axes and extrusion speed, ensuring consistent material deposition throughout the print. Experimental results also confirm the system’s ability to produce smooth, continuous profiles and demonstrate that the extrusion motor speed can remain constant while other axis drivers are adjusted according to the location to be printed. The study concludes that polar kinematics combined with piston-based extrusion is a viable and efficient solution for desktop-scale ceramic additive manufacturing, offering a foundation for future development in architectural, mechanical, and functional clay printing.