Regular polygon area to perimeter ratio!

A $n$ -sided regular polygon is made up of $n$ isosceles triangles with the side of polygon as base and equal sides of length $R$ . The angle between the two equal sides is the central angle of the polygon which is $\frac {2\pi}n$ (see figure).

Then, the area of the polygon $A$ is $n$ times the area of the triangle $A_\triangle$ or $A = n A_\triangle = n R^2 \sin \frac \pi n \cos \frac \pi n$ .

The perimeter of the polygon $P = 2 n R \sin \frac \pi n$ .

Therefore, $\dfrac AP = \dfrac {n R^2 \sin \frac \pi n \cos \frac \pi n}{2 n R \sin \frac \pi n} = \dfrac R2 \cos \dfrac \pi n$ . Then $k+m = 1+2 = \boxed{3}$ .