Question

If the diameters of the concentric circles as shown in the given figure are in the ratio 1 : 2 : 3 then find the ratio of the areas of three regions.

Answer 1

Given: The diameters of three concentric circles are in the ratio 1 : 2 : 3.

Step-wise calculation:

1. Let the diameters be k, 2k, 3k.

Then the radii are `k/2, k, (3k)/2`.

2. Area of inner circle

`A_1 = π (k/2)^2` 

= `(πk^2)/4`

3. Area of middle annular region

A₂ = π(k)² – A₁ 

= `πk^2 - (πk^2)/4`

= `(3πk^2)/4`

4. Area of outer annular region

`A_3 = π((3k)/2)^2 - πk^2` 

= `(9πk^2)/4 - πk^2`

= `(5πk^2)/4`

5. Ratio A₁ : A₂ : A₃

= `((πk^2)/4) : ((3πk^2)/4) : ((5πk^2)/4)`

= 1 : 3 : 5

The areas of the three regions are in the ratio 1 : 3 : 5.