The area of the circle and the area of a regular polygon inscribed the circle of n sides and of perimeter equal to that of the circle are in the ratio of

Explanation

Area of the circle` (A_1) = pir^2`

It is given that perimeter of the circle and the polygon are equal.

`:. na = 2pir`...[Here, ` a` is the side of the polygon]

`=>a = (2pir)/n`

Now, Area of the polygon `(A_2) = 1/4na^2cot(pi/n)`

`=>A_2 = 1/4n(4pi^2r^2)/n^2cot(pi/n)`

`=>A_2 = (pi^2r^2)/ncot(pi/n)`

`:. A_1/A_2 = (pir^2)/((pi^2r^2)/ncot(pi/n)) = tan(pi/n)/(pi/n)`

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