The area of the circle and the area of an regular polygon of sides of perimeter equal to that of the circle are in the ratio of
- $\tan { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
- $\cot { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
- $\cos{ \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
- $\sin { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
For a circle of radius r, area is pi*r^2 and perimeter is 2*pi*r. For a regular polygon with n sides of total perimeter 2*pi*r, each side length is 2*pi*r/n. The area of such a polygon is (n * s^2) / (4 * tan(pi/n)), where s is the side length. Substituting s, the ratio simplifies to tan(pi/n) / (pi/n).
Let the radius of the circle be r, making its circumference and the perimeter of the polygon 2*pi*r. If the regular polygon has n sides, each side is 2*pi*r/n. The area of the circle is pi*r^2. The area of a regular polygon with n sides of length s is (n*s^2)/(4*tan(pi/n)); substituting s = 2*pi*r/n gives the polygon area as (n*(2*pi*r/n)^2)/(4*tan(pi/n)) = (pi^2*r^2)/(n*tan(pi/n)). The ratio of the circle's area to the polygon's area is (pi*r^2) / ((pi^2*r^2)/(n*tan(pi/n))), which simplifies to tan(pi/n) : pi/n.