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

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Solution

Let $r_{1}$ and $r_{2}$ be the radii of two given circles and $C_{1}$ and $C_{2}$ be their circumference.

Then, $C_{1} : C_{2} = 2 : 3$

$\Rightarrow 2 π r_{1} : 2 π r_{2} = 2 : 3$

$\Rightarrow r_{1} : r_{2} = 2 : 3$

$\Rightarrow r_{1}^{2} : r_{2}^{2} = 4 : 9$

$\Rightarrow π r_{1}^{2} : π r_{2}^{2} = 4 : 9$

Hence, ratio of areas is 4:9.

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