In this article, we examine the smallest enclosing circle problem, which is one of the most important problems in facility location and computational geometry that has numerous applications in various fields such as machine vision, computer graphics, collision detection, and robotics. The smallest enclosing circle problem seeks to find the smallest circle that can enclose a given set of n points in a two-dimensional plane. In this article, we present a new 1.41-approximation algorithm with time complexity of O(n) for solving the smallest enclosing circle problem that guarantees a solution that is at most 1.41 times of the optimal enclosing circle. To the best of the authors' knowledge, this result is the best fixed approximation factor in a linear time approximation algorithm for this problem. The results of the experimental implementation of the proposed algorithm compared to the exact algorithm on random data with different distributions show that the proposed approximation algorithm provides a close to optimal result in most cases, and its average optimality ratio for the data under study is 1.095.