In a \(n\) side regular polygon, there are exactly \(n\) similar triangles whose base is on one side of the polygon. The sum of the area of these triangles \((A_t)\) is the area of the regular polygon, \(A_p\).

\(A_p = nA_t\)

The angle of the triangle whose vertex is at the center of the circle can be calculated by \(\frac{360}{n}\). Thus, \(\theta = \frac{360}{2n} = \frac{180}{n}\).

The area of the triangle, \(A_t = \frac{1}{2} b r\)

\(\tan \theta = \frac{b}{2r}\)

\(b = 2r \tan \theta\)

\(A_t = r^2 \tan \theta\)

Therefore, \(A_p = nr^2\tan \theta\) but, \(\theta = \frac{180}{n}\), thus \(A_p = nr^2\tan \frac{180}{n}\)