# ABCD is a square. The midpoints of the sides are joined to form a square A1,B1,C1,D1. The same way we form another square A2,B2,C2,D2 from A1,B1,C1,D1. This process of forming squares is continued. Question: The ratio of the areas of the circum circle and area of the square is ? A. Depends on the radius of the circle. B. Independent of the radius of circle. If A why ? If B why?

420 Points
13 years ago

Dear Vivek

Question is not clear.i.e. ratio of which circumcircle and which square beacause as mentioned in the question there are many squares ABCD,A1B1C1D1, A2B2C2D2..etc.

All the best.

AKASH GOYAL

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509 Points
13 years ago

let the side of outermost square is a....

after drawing the diagram it is clear that  radius of circumcircle is 1/21/2 times the side of square ....

side of inner square is 1/21/2 times side of outer square.....

now if the side of outermost square is a then we have

a1,a2,a3,a4.......        =     a        ,   a/21/2 ,       a/2   ,       a/2.21/2  and so on...

r1,r2,r3,r4........         = a/21/2   ,      a/2    ,      a/2.21/2  ,     a/4  and so on

now

area's of circumcircle is ,

Ac = pi(r12 + r22 + r32  ............infinity)

=(pi)a2 (1/2 +1/4 +1/8 +1/16 .....infinity)                    (sum of infinite GP is a/a-r)

=(pi)a2 ............1

area's of square is ,

As =a2 (1 + 1/2 + 1/4 + 1/8 ...........infinity)

As = 2a2 ...............2

dividing 1 and 2

eq1/eq2=pi/2            independent of radius of circle...

hence option B is correct