The circumference of two circles are in the ratio 2 : 3. Find the ratio of their areas.

Given:

The circumference of two circles are in the ratio 2 : 3.

To do:

We have to find the ratio of their areas.

Solution:

Let the radius of the two circles be $r_1$ and $r_2$.

We know that,

Circumference of a circle of radius $r=2 \pi r$

Area of a circle of radius $r=\pi r^2$

Therefore,

Ratio of the circumference of the circles $=2 \times \frac{22}{7} \times r_1:2 \times \frac{22}{7} \times r_2$

$r_1:r_2=2:3$

Ratio of the areas of the circles $=\frac{22}{7} \times(r_1)^{2}:\frac{22}{7} \times(r_2)^{2}$

$=r_1^2:r_2^2$

$=(\frac{r_1}{r_2})^2$

$=(\frac{2}{3})^2$

$=\frac{4}{9}$

The ratio of the areas of the circles is $4:9$.