If the perimeter of a circle is equal to the perimeter of a regular polygon of 'n' sides, then their areas are in the ratio:
- $\displaystyle \tan \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
- $\displaystyle \cos \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
- $\displaystyle \sin \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
- $\displaystyle \cot \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
For a circle and a regular n-sided polygon with equal perimeter P, the radius of the circle is r = P/(2*pi). The apothem of the polygon is a = P/(2*n*tan(pi/n)). The ratio of areas is (pi * r^2) / (n * s * a / 2) = (pi * (P/2pi)^2) / (P * a / 2) = tan(pi/n) / (pi/n).
Let the common perimeter be P, making the circle's radius r equal to P divided by 2*pi and the polygon's side length s equal to P divided by n. The area of the circle is pi times r squared, which simplifies to P squared divided by 4*pi. The area of a regular polygon is given by n times s squared divided by 4 times the tangent of pi divided by n, which simplifies to P squared divided by 4 times n times the tangent of pi divided by n. The ratio of the circle's area to the polygon's area is n times the tangent of pi divided by n divided by pi, which simplifies to the tangent of pi divided by n divided by pi divided by n.