In all the pictures, below, centres of the circles are on the same line. In the first two pictures, the small circles are of the same diameter.

Prove that in all pictures, the perimeters of the large circle is the sum of the perimeters of the small circles.

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 3^rd 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.