A Geometric Solution to the Isoperimetric Problem and its Quantitative Inequalities

This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons converge to the circle in area efficiency as the number of sides increases. The paper also extends these ideas to irregular polygons using average apothem and circumradius, defines "wasted area," and introduces several compactness and smoothness indices, offering a unified and quantitative view of isoperimetric inequalities