Polygons

A polygon is a shape made of segments connected to each other end-to-end. The simplest polygon is the triangle. The name of a polygon indicates the number of its sides or interior angles.

Each point where two segments are connected is called a vertex. Each two sides connected at a vertex are called adjacent sides. Each two vertices (plural of vertex) on the same side are called adjacent vertices.

An angle inside a polygon between two of its sides is called an interior angle, while an angle between a side and the extension of the adjacent side is called an exterior angle. A polygon with \(n\) sides (and angles) is called an \(n\)-gon. A diagonal is a segment that connects non-adjacent vertices. The number of diagonals of an \(n\)-gon is \[\text{Number of diagonals}=\dfrac{n(n-3)}{2}\]

An \(n\)-gon can be divided into \((n-2)\) triangles by diagonals from one and the same vertex. Therefore, the sum of the interior angles of an \(n\)-gon is \[\text{Sum of interior angles}=(n-2)180^\circ\]

A regular polygon is one that has equal sides. It follows that it also has equal interior angles. The interior angle of a regular \(n\)-gon is \[\text{Interior angle of a regular }n\text{-gon}=\dfrac{(n-2)180^\circ}{n}\]