In this paper, a method for mobile robot positioning using range data is investigated. A ranging sensor is used to collect range data within an environment modeled as a closed polygon, and a compass is available to provide the ranging sensor with direction. The problem of position uncertainty is defined as the existence of multiple position solutions when angle readings are noiseless. When the compass is not ideal, the problem is further complicated by now having imprecise multiple solutions. Both cases are solved by introducing a function defined as the region entropy, which provides us with the ranging angle directions that minimize the uncertainty in position estimation. A least square position estimation method is then proposed. Finally, an example is used to illustrate the technique described above.