A Geometric Solution to the Isoperimetric Problem and its Quantitative Inequalities

This paper introduces a geometric solution to the classical isoperimetric problem by analysing the area efficiency of n-sided regular polygons through a novel apothem-hypotenuse ratio framework. Many new formulas are derived to quantify polygonal efficiency based on apothem, hypotenuse, and perimeter, demonstrating how regular polygons approach the optimal area-enclosing efficiency of the circle as the number of sides increases. This paper derives several bounded efficiency metrics for regular polygons. This approach provides fresh insight into quantitative isoperimetric inequalities through direct geometric reasoning, with accuracy confirmed by both analytical derivation and numerical testing.