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

Options

A. `tan((pi)/(n)): (pi)/(n)`
B. `cos ((pi)/(n)) : (pi)/(n)`
C. `sin.(pi)/(n): (pi)/(n)`
D. `cot((pi)/(n)): (pi)/(n)`

Answer

A

Explanation

Let r be the radius of the circle and `A_(1)` be its area. Then `A_(1) = pir^(2)`. Since the perimeter of the circle is same as the perimeter of a regular polygon of n sides, we have `2pi r = na`, Where a is the length of one side of the regular polygon. Thus,
`a = (2pi r)/(n)`
Let `A_(2)` be the area of the polygon. Then,
`A_(2) = (1)/(4) na^(2) cot ((pi)/(n)) = (pi^(2) r^(2))/(n) cot ((pi)/(n))`
`rArr A_(1) : A_(2) = pi r^(2) : (pi^(2) r^(2))/(n) cot ((pi)/(n)) = tn ((pi)/(n)) : (pi)/(n)`