In this letter, we present algorithms to estimate contact locations, external forces, and joint torques using skin, i.e., distributed tactile sensors, kinematic sensors, and a single Inertial Measurement Unit (IMU) without the need for force-torque sensors. Distributed tactile sensors are an array of discrete tactile sensors (taxels) with gaps between them. To cope with the gaps, we use interpolation techniques to improve accuracy of the measurements in the skin. Then, we estimate the external force. Finally, we use this information combined with the contact location obtained from the skin to estimate the joint torques. Validation was performed using the iCub humanoid robot. Results show a performance comparable to the joint torque estimation using expensive force-torque sensors. Since the skin is less expensive and does not require modification of the internal structure of the robot, using the skin is an interesting solution.