The purpose of this paper is to discuss length estimation based on digitized curves. Information on a curve in the Euclidean plane is lost after digitization. Higher resolution supports a convergence of a digital image towards the original curve with respect to Hausdorff metric. No matter how high resolution is assumed, it is impossible to know the length of an original curve exactly. In image analysis we estimate the length of a curve in the Euclidean plane based on an approximation. An approximate polygon converges to the original curve with an increase of resolution. Several approximation methods have been proposed so far. This paper proposes a new approximation method which generates polygonal curves closer (in the sense of Hausdorff metric) in general to its original curves than any of the previously known methods and discusses its relevance for length estimation by proving a Convergence Theorem.