Circular statistics has found substantial application in science and engineering. One of the fundamental problems in circular statistics is that of estimating the mean direction of a circular random variable from a number of observations. The standard approach in the literature is called the sample circular mean and its asymptotic properties are well known. It can also be computed efficiently in a number of arithmetic operations that is linear in the number of observations. In this paper we consider an alternative estimator called the sample intrinsic mean that is based on minimizing squared arc length. We show how this estimator can be computed efficiently in a linear number of operations using an algorithm from algebraic number theory and we derive its asymptotic properties. In some scenarios the sample circular mean and the sample intrinsic mean are estimators of the same quantity and can therefore be compared. We show both theoretically and by simulation that in some of these scenarios the sample intrinsic mean is statistically more accurate than the sample circular mean. As such the results in this paper potentially have implications for the wide variety of fields in science, engineering and statistics that currently use the sample circular mean.