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

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

Let’s assume the radius of two circles ${{C}_{1}}$ and ${{C}_{2}}$ be ${{r}_{1}}$ and ${{r}_{2}}$

We all know that, Circumference of a circle (C) $=2\pi r$

And their circumference will be $2\pi {{r}_{1}}$ and $2\pi {{r}_{2}}$ .

Then, their ratio is $={{r}_{1}}:{{r}_{2}}$

Given in the question, circumference of two circles is in a ratio of $2:3$

${{r}_{1}}:{{r}_{2}}=2:3$

Now, the ratios of their areas is given as

$=\pi r_{1}^{2}:\pi r_{2}^{2}$

$={{\left( \frac{r1}{r2} \right)}^{2}}$

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

$=\frac{9}{16}$

$=\frac{4}{9}$

Thus, ratio of their areas $=4:9$ .

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