A regular polygon has all sides equal #a#. Each side is observed from the center #O# at the same angle #phi#. The polygon area is covered by as many isosceles triangles with vertices centered, as sides it has. A regular polygon has a circumscribed circle with radius #r#. Considering now the radius #r# as the side of isosceles triangle with vertice at the center and #a# as third side we have the area #s#

#s = 1/2a r cos(phi/2)#

if we have #n# sides then #phi=(2pi)/n#

and the polygon area is given by

#S = n s = (nar)/2cos(pi/n)#

but #a = 2rsin(phi/2)=2rsin(pi/n)#

now substituting #r = a/(2sin(pi/n))# into the #S# relationship

#S=(n a^2)/4cot(pi/n)#