Let S be a set of points in the plane, CH be the convex hull of S, ℘(S) be the set of all simple polygons crossing S, γ P be the maximum angle of polygon P ∈ ℘(S) and θ = min P ∈℘(S) γ P . In this paper, we prove that θ ≤ 2π − 2π r.m such that m and r are the number of edges and internal points of CH, respectively. We also introduce an innovative polynomial time algorithm to construct a polygon with the said upper bound on its angles. Constructing a simple polygon with angular constraint on a given set of points in the plane is highly applicable to the fields of robotics, path planning, image processing, GIS, etc. Moreover, we improve our upper bound on θ and prove that this is tight.