Three-dimensional (3D) braided composites have significant potential for use in engineering structural materials. However, conventional 3D braiding machines are insufficient for designing composites with complex geometries. This paper proposes a programmable design methodology for 3D rotary braiding machines using circle-cutting and combination strategies. By introducing varying numbers of incisions on the circle, a diverse range of horn gears can be designed. Different combinations of these cut-circles allow the horn gears to be assembled into various 3D rotary braiders. The parametric equation for the braider plate is derived, showing that a combination strategy involving two cut-circles is feasible for braider design, whereas integrating three cut-circles simultaneously is impossible for a single machine. The construction of an automatic 6-3 type 3D braiding machine demonstrates the effectiveness of the proposed design strategy. This flexible braider design approach provides a practical solution for producing 3D braided composites with complex geometries.