PERIMETER AND AREA OF REGULAR POLYGONS AND PTOLEMY’S THEOREM

Jumaev Asilbek Shuhrat ugli

Authors

Jumaev Asilbek Shuhrat ugli Science teacher at the Academic Lyceum of Termiz State University of Engineering and Agrotechnology Author

Keywords:

regular polygon, perimeter, area, apothem, cyclic quadrilateral, Ptolemy’s Theorem, Euclidean geometry.

Abstract

This paper examines the geometric properties of regular polygons, focusing on the derivation and application of formulas for perimeter and area. It explains how a regular polygon, defined by equal sides and equal angles, can be analyzed using both algebraic and trigonometric approaches. The study presents the standard formulas for the perimeter

and the area , showing how these expressions are derived through decomposition into congruent triangles.

In addition, the paper investigates Ptolemy’s Theorem, a classical result in Euclidean geometry that establishes a relationship between the sides and diagonals of a cyclic quadrilateral. The theorem is stated and interpreted geometrically, emphasizing its significance in solving complex geometric problems and proving trigonometric identities. The connection between regular polygons inscribed in a circle and Ptolemy’s Theorem is also discussed, highlighting its broader mathematical relevance.

The research demonstrates that both topics are fundamental in understanding spatial structures, symmetry, and mathematical reasoning. These concepts are widely applicable in fields such as architecture, engineering, computer graphics, and mathematical modeling.