How to calculate the largest area of a rectangle inscribed in a circle with a radius of 13?

1 Answer
Feb 16, 2018

drawn

Let us consider that #ABCD# is the rectangle inscribed in a circle of radius #13# unit. As semicircular angle is right angle, diagonal #AC# will be the diameter of the circle. So #AC=26# unit.

Now let the #angleACB=theta#

So side #BC=ACcostheta# and side #AB=ACsintheta#

Now area #(S)#of the inscribed rectangle will be given by
#S=ABxxBC=ACsinthetaxxACcostheta=AC^2sinthetacostheta#

#=>S=26^2sintheta costheta=2xx13^2sin2theta#

Now area #S # will be maximum when #sin2theta# becomes maximum i.e #sin2theta=1=>theta=45^@#

Hence maximum area #S_"max"=2xx13^2=338# squnit

As #theta=45^@#, the rectangle becomes a square which is treated as a special kind of rectangle.