Answer :

Let R = radius of larger circle in all figures.


If r = radius of circle


Circumference =


Perimeter of large circle = …(eq)1


In first figure,


Diameter of larger circle = 2 × Diameter of smaller circle(as seen from figure)


Hence,


Diameter of smaller circle =


Radius of smaller circle = (Radius = )


Let r = Radius of smaller circle


r =


Perimeter of a smaller circle = 2 × π × r


= 2 × π × = π × R


Perimeter of 2 smaller circles = 2 × (π × R) = …(eq)1


Hence, proved.


In second figure,


Diameter of larger circle = 3 × Diameter of smaller circle(as seen from figure)


Hence,


Diameter of smaller circle =


Radius of smaller circle = (Radius = )


Let r = Radius of smaller circle


r =


Perimeter of a smaller circle = 2 × π × r


= 2 × π × R


Perimeter of 3 smaller circles = 3 × ( × π × R) = …(eq)1


Hence, proved.


In 3rd figure,


There are 3 small circles.


Right half has radius =


Perimeter of right half = 2 × π × = π × R …(eq)2


The left half’s radius is divided in the ratio 1:2


Hence, radius of smallest circle =


Hence, radius of middle circle =


Perimeter of smallest circle = 2 × π × …(eq)3


Perimeter of middle circle = …(eq)4


Sum of perimeter of all circles = (eq)2 + (eq)3 + (eq)4


= (π × R) + (π × ) + (2 × π × )


= π × R …(same as (eq)1)


Hence, proved.


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