In this paper, we present a general version 2D Shape Equipartition Problem (2D-SEP) under minimum boundary length. The goal of this problem is to obtain a segmentation into N equal area segments (regions), where the number of segments (N) is given by the user, under the constraint that the boundaries between the segments have a minimum length. 2D-SEP is defined without any assumption or prior knowledge of the object structure and the location of the segments. In this work, we define the 2D-SEP and we propose a fast region growing based method that solves the general version of 2D-SEP problem. Additionally, we study the special case of the 2D-SEP in which the intrinsic boundaries are line segments, proving that it has at least one solution in convex shapes and presenting a sequential selection method that efficiently solves the problem. The quantitative results obtained on more than 2,800 2D shapes included in two standard datasets quantify the performance of the proposed methods.