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If #I_n# be the area of n-sided regular polygon inscribed in a circle of unit radius and #O_n# be the area of the regular polygon circumscribing the given circle prove that

#I_n=((O_n)/2)[1+sqrt(1-((2I_n)/n)^2)]#

1 Answer
P dilip_k
Jun 6, 2018

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Given that an n-sided regular polygon is inscribed in a circle of unit radius i.e. #R=1#. Another regular polygon of n sides also circumscribes this until circle. Let circum radius of larger polygon be #R'#

Let #theta# be the angle subtended by each side of a an n-sided regular polygon at the center of it i.e at the circumcenter of the regular polygon.

Now area of the inner smaller polygon will be

#I_n =n*1/2R^2sintheta=n/2sintheta#

So #sintheta=(2I_n)/n#

Now #R=R'cos(theta/2)#

#=>R'=1/cos(theta/2)#

Again the area of the larger outer polygon will be

#O_n=n/2(R')^2sintheta#

So
#I_n/O_n=1/(R')^2#

#=>I_n/O_n=1/((1/cos(theta/2))^2#

#=>I_n=O_n/2*2cos^2(theta/2)#

#=>I_n=O_n/2(1+costheta)#

#=>I_n=O_n/2(1+sqrt(1-sin^2theta))#

#=>I_n=O_n/2[1+sqrt(1-((2I_n)/n)^2)]#

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