The ratio of the area of a regular polygon of `n` sides inscribed in a circle to that of the polygon of same number of sides circumscribing the same is 3:4. Then the value of `n` is

Explanation

Area of the polygon inscribed in a circle, `(S_1) = 1/2na^2sin((2pi)/n)`

Area of the polygon circumscribing the same circle, `(S_2) = na^2tan(pi/n)`

It is given that,

`S_1/S_2 = 3/4`

`:. (1/2na^2sin((2pi)/n))/(na^2tan(pi/n)) = 3/4`

`=>(1/2sin((2pi)/n))/(tan(pi/n)) = 3/4`

`=>(1/2*2sin(pi/n)cos(pi/n))/(sin(pi/n)/cos(pi/n)) = 3/4`

`=>cos^2(pi/n) = 3/4`

...