This paper presents a distributed fermat-point range estimation strategy, which is important in the moving sensor localization applications. The fermat-point is defined as a point which minimizes the sum of distances from three sensors inside a triangle. This point is indeed at the triangle's center of gravity. We solve the problems of large errors and poor performance in the bounding box algorithm. We obtain two results by performance analysis for a deployed environment with 200 sensor nodes. First, when the number of sensor nodes is below 150, the mean error decreases rapidly as the node density increases, and when the number of sensor nodes exceeds 170, the mean error stays below 1%. Second, when the number of beacon nodes is below 60, the normal nodes do not have sufficient number of accurate beacon nodes to help them estimate their locations. However, when the number of beacon nodes exceeds 60, the mean error changes slightly. Simulation results indicated that the proposed algorithm for sensor position estimation is more accurate than existing algorithms and improves on existing bounding box strategies.