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If In is the area of n-s i d e d regular...

If `I_n` is the area of `n-s i d e d` regular polygon inscribed in a circle of unit radius and `O_n` be the area of the polygon circumscribing the given circle, prove that `I_n=(O_n)/2(sqrt(1+((2I_n)/n)^2))`

Text Solution

Verified by Experts

We know that, `I_(n)=n/2r^(2)sin""(2pi)/n`
[since, `I_(n)` is area of regular polygon ]
`rArr" "(2I_(n))/n=sin""(2n)/n" "[becauser=1]....(i)`
`"and "O_(n)=nr^(2)tan""pi/n`
[since, `O_(n)` is area of circumscribin polygon]
`O_(n)/n=tan""(pi)/n" ...(ii)"`
On dividing Eq. (i) by Eq. (ii), we get
`(2I_(n))/O_(n)=(sin""(2pi)/n)/(tan""pi/n)`
`rArr" "I_(n)/O_(n)=cos^(2)""pi/n=(1+cos""(2pi)/n)/2`
`:." "I_(n)/O_(n)=(1+sqrt(1-(2I_(n)//n)^(2)))/2" "["from Eq. (i)]`
`rArr" "I_(n)=O_(n)/2(1+sqrt(1-(2I_(n)//n)^(2)))`
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