Estimating the distance between two points is of fundamental concern. This paper investigates some statistical properties of three estimators of the distance between two points on a plane. The results of several theoretical comparisons of the performance of the estimators assuming a large sample size are given. Also given is the comparison of the performance of the estimators using simulation when the sample size is small. These comparisons suggest that the estimator of choice is not the most "natural" estimator in this situation. Although the discussion is given in the framework of the plane, the results are readily extended to high dimensional spaces.