Ratio of the Sides Between a Regular Polygon and an Interior Polygon Formed by its Non-Opposite Diagonal Lines..

This article investigates the geometric relationship between a regular polygon (the exterior polygon) and a smaller, similar interior polygon formed by intersecting non-opposite diagonals of the original polygon. It distinguishes two cases based on whether the polygon has an odd or even number of sides and introduces the concept of "levels of compression" (denoted by an integer k) to describe successive interior polygons formed by these diagonals. Using properties of cyclic polygons and the Law of Sines, the article derives a general formula expressing the ratio of the side length of the interior polygon to that of the exterior polygon in terms of n (the number of sides) and k. The formula is validated through examples involving polygons from triangles to octagons, with measured values closely matching theoretical predictions, and potential applications are suggested in fields such as origami and engineering for material compression.