This paper is devoted to an octagonal cut algorithm and a hexadecagonal cut algorithm for finding the convex hull of 𝑛 points in R^2, where some octagon and hexadecagon are used for discarding most of the given points interior to these polygons. In this way, the scope of the given problem can be reduced significantly. In particular, the convex hull of 𝑛 points distributed 𝑏least -𝑏most -boundedly in some rectangle can be determined with the complexity 𝑂(𝑛). Computational experiments demonstrate that our algorithms outperform the Quickhull algorithm by a significant factor of up to 47 times when applied to the tested data sets. The speedup compared to the CGAL library is even more pronounced.