Consider the calculations for the area of regular polygons with \(n = 3,\;4,\;5\) and \(6\) sides inscribed within a circle of radius \(r\). In each case, the polygon can be divided into \(2n\) identical right triangles each with an area of \(\frac{{bh}}{2}\) where \(b\) and \(h\) are, respectively, the base and height of the right triangle.The calculations clearly show that the area of each regular polygon is \(A = n{r^2}\sin \left( {\frac{{\rm{\pi }}}{n}} \right)\cos \left( {\frac{{\rm{\pi }}}{n}} \right)\).