We know that total number of vertices in a polygon having \(n\) sides, is \(n\). Hence total number of lines which can be formed by joining any two points = \(^n{{\rm{C}}_2}\), and of these lines n are sides of polygons, hence number of diagonals = \(^n{{\rm{C}}_2} - n\) = \(\frac{{n(n - 3)}}{2}\)

Consider a convex hexagon, there are 6 vertices and 6 sides. Total number of line segments that can be formed by connecting these vertices = \(^6{{\rm{C}}_2} = 15\), but out of these 15 line segments, 6 are sides of the hexagon, and the remaining 9 are the diagonals, shown in green colour.

Example 01:  Number of diagonals in a convex polygon is four times the number of sides. Find the number of sides in the polygon.