Polygon

Polygon is a plane figure in geometry, defined as a closed chain of straight line segments joined end to end to form a simple polygon, typically lying in a single plane. A polygon is convex if every line segment between two points inside the polygon lies inside the polygon; if not, it is concave. A polygon with n sides has n vertices. The interior angle sum of an n-sided polygon is (n − 2) × 180 degrees, and the exterior angles sum to 360 degrees. A regular polygon has all sides and all interior angles equal.

Diagonals connect nonadjacent vertices; a polygon with n vertices has n(n − 3)/2 diagonals. The area of a polygon can be computed in several ways, including the shoelace formula for coordinate representations, or, for a regular n-gon with side length s, area = (n s^2) / (4 tan(π/n)). For a regular polygon with circumradius R, area = (n/2) R^2 sin(2π/n).

Polygons can be described by vertex coordinates or edge vectors. They are fundamental in mathematics and appear in computer graphics, geographic information systems, engineering, and architecture. Special cases include triangles (3 sides) and quadrilaterals (4 sides).

In technology, Polygon also refers to a blockchain platform that provides layer-2 scaling solutions for Ethereum, originally launched as Matic Network and rebranded to Polygon. It aims to offer faster and cheaper transactions and support for decentralized applications.