The circumference of two circles are in

Question:

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

Solution:

Let the radius of two circles beandrespectively. Then their circumferences are and respectively and their areas are and respectively.

It is given that,

$\frac{C_{1}}{C_{2}}=\frac{2}{3}$

$\frac{2 \pi r_{1}}{2 \pi r_{2}}=\frac{2}{3}$

$\frac{r_{1}}{r_{2}}=\frac{2}{3}$

Now we will calculate the ratio of their areas,

$\frac{A_{1}}{A_{2}}=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}$

$=\frac{r_{1}^{2}}{r_{2}^{2}}$

$=\left(\frac{r_{1}}{r_{2}}\right)^{2}$

Substituting the value of $\frac{r_{1}}{r_{2}}$,

$\frac{A_{1}}{A_{2}}=\left(\frac{2}{3}\right)^{2}$

$=\frac{4}{9}$

Hence the ratio of their Areas is $4: 9$.

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