If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

Consider the figure shown below where O is centre of circle, join BC which passes through O, let the side of square be ‘a’ and radius of circle be ‘r’.


Now we know OB and OC are radius of circle


So, OB = OC = r



Consider ∆BDC right angled at D




And we know BC = OC + OB


BC = 2r and BD = DC = a (put these values in eqn1)


(2r)2 = a2 + a2


4r2 = 2a2




Area of inscribed square = side × side


Areaa of inscribed square = a × a


Area of inscribed square = a2 eqn3


Area of circumscribing circle = πR2 where R = radius of circle


Area of circumscribing circle = πr2 eqn4



Put the values from equation 3 & 4 in above equation




(from eqn 2)



So, Ratio is π : 2


The ratio is π:2


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