Gauss Circle Problem Solving and Extending in 3D Shape Using Pick’s Theorem

In this study, a new solution to the Gauss Circle Problem was presented using Pick’s Theorem. The simplest polygon containing both grid points inside the circle and on the circumference was drawn, and the result was derived by applying Pick’s Theorem to the polygon instead of the circle. He analyzed the existing solution method and inductively proved using Python that the conclusion obtained through the new solution method was within the error range known through the existing solution method, revealing that the new solution method was valid. A two-dimensional circle was expanded into a three-dimensional sphere, and a new solution method was applied to three-dimensional space using Reeve’s Theorem, which extends Pick’s Theorem to three dimensions.