Polygon
Key Formulas
| Property | Statement |
|---|---|
| Interior angle sum | (n−2)·180° |
| Regular interior angle | (n−2)·180°/n |
| Diagonals | n(n−3)/2 |
Examples
Example 1. Sum of interior angles of a hexagon.
Solution. (6−2)·180°=720°.
In Depth
A polygon is a closed plane figure bounded by straight line segments. Regular polygons — all sides and angles equal — have been studied since antiquity. The ancient Greeks knew how to construct regular polygons with 3, 4, 5, 6, and 15 sides using compass and straightedge. Gauss proved at age 18 that a regular 17-gon is constructible, and characterized all constructible regular polygons.
The interior angle sum of an \(n\)-gon is \((n-2)\cdot180°\). For a regular \(n\)-gon, each interior angle is \((n-2)\cdot180°/n\). As \(n\to\infty\), the regular \(n\)-gon approaches a circle and each interior angle approaches 180°.
The area of a regular \(n\)-gon with side length \(s\) is \(A=\frac{ns^2}{4}\cot(\pi/n)\). Polygons tile the plane only for \(n=3,4,6\) (triangles, squares, hexagons) — a fact with implications for crystallography and architecture.
In computational geometry, polygon algorithms (convex hull, point-in-polygon, polygon clipping) are fundamental to GIS, robotics, and computer graphics. The shoelace formula computes the area of any polygon from its vertex coordinates: \(A=\frac{1}{2}|\sum_{i}(x_iy_{i+1}-x_{i+1}y_i)|\).
Further Reading & Context
The study of polygon connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.