Question

The circumference of two circles are in the ratio of 2:3. The ratio of their areas will be? ​

Answer 1

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Let the radius of smaller circle be r and bigger circle be R

Given

The ratio of their circumferences is 2:3

$\implies \: \sf{ \frac{ \cancel{2 \pi} \: r}{ \cancel{2 \pi} \: R} = \frac{2}{3} } \\ \\ \implies \: \boxed{\sf{r \colon R = 2 \colon \: 3}}$

Implies,(r,R) = (2,3)

To find

Ratio of their areas

Let x and y be the ratio of their areas

Now,

$\sf{ \frac{x}{y} = \frac{ \cancel{\pi} r {}^{2} }{ \cancel{\pi} {R}^{2} } } \\ \\ \\ \implies \: \sf{ \frac{x}{y} = \frac{2 {}^{2} }{3 {}^{2} } } \\ \\ \huge{\implies \: \boxed{\sf{x : y = 4 : 9}}}$

Hence,the ratio of their areas is 4 : 9

Answer 2

Answer:

Let the radius of two circles be r and R.

Circumference of circle = 2πr

Given :

Circumference of the two circles are in ratio = 2:3

$\: \: \: \: \: \: \: \: \: 2\pi r : 2\pi R = 2 : 3\\ = > \frac{2\pi r}{2\pi R} = \frac{2}{3} \\ = > \frac{r}{R} = \frac{2}{3}$

To find ratio of areas :

Let areas of the two circles be a and A.

$Area \: of \: circle = \pi {r}^{2}$

$\: \: \: \: \: \: \: \: \frac{a}{ A} = \frac{\pi {r}^{2} }{\pi {R}^{2} } \\ = > \frac{a}{A} = \frac{ {2}^{2} }{ {3}^{2} } \\ = > \frac{a}{A} = \frac{4}{9} \\ = > a : A = 4 : 9$

Ratio of areas = 4 : 9