If the area of the circle is `A_1` and the area of the regular pentagon inscribed in the circle is `A_2,` then find the ratio `(A_1)/(A_2)dot`

Explanation

In `Delta OAB, OA = OB = r`
and `angle AOB = (360^(@))/(5) = 72^(@)`
`:.` Area of `DeltaAOB = (1)/(2) r xx r sin 72^(@)`
`= (1)/(2) r^(2) cos 18^(@)`
`:.` Area of pentagon `(A_(2)) = 5 ("Area of " DeltaAOB)`
`= 5((1)/(2) r^(2) cos 18^(@))` (i)
Also, Area of the circle `(A_(1)) = pi r^(2)` (ii)
Hence, `(A_(1))/(A_(2)) = (pi r^(2))/((5)/(2) r^(2) cos 18^(@)) = (2pi)/(5) sec ((pi)/(10))`
[from Eqs. (i) and (ii)]