With the proliferation of haptic devices, there has been significant research toward realistic haptic rendering of virtual environments. Currently, the majority of haptic devices provide only three degrees-of-freedom (3 DOF), but 5 and 6 DOF devices are becoming more common. This paper presents a method of using existing 3 DOF rendering techniques to produce forces for a 5 or 6 DOF device when true 6 DOF rendering is unavailable. Point-force 3 DOF rendering that is most commonly used in commercial haptic applications is simple and fast for basic haptic rendering; however, this creates unrealistic haptic effects when an avatar other than a point is considered. As a solution, a 6 DOF force and torque algorithm is presented based on multiple contact points. Using the multi-point torque rendering approach, forces are rendered in three linear dimensions using existing 3 DOF algorithms, and up to three additional torque degrees-of-freedom are calculated based on the forces on multiple points. This capability enhances haptic realism without modifying the legacy rendering. Finally, this algorithm is demonstrated using a 5 DOF haptic interface. The results from test observations suggest that the 5 DOF rendering algorithm functions as expected.