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The circumferences of two circles are in the ratio $3 : 4$ . The ratio of their areas is,

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3 : 4

4 : 3

9 : 16

16 : 9

answer is C.

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Detailed Solution

It is given that the circumferences of two circles are in the ratio

3 : 4

. We need to find the ratio of their areas.
Let the radius of the two circles be

r

and

R

respectively.
We know that the circumference of a circle is

2 πr

units, where

r

is the radius of the circle.
Therefore,
Circumference of the circle with radius

r

2 πr

units.
Circumference of the circle with radius

R

2 πR

units.
By data,

\frac{2 πr}{2 πR} = \frac{3}{4}

\Rightarrow \frac{r}{R} = \frac{3}{4}

— (1)
We know that the area of a circle is

π r^{2}

uni t^{2}

, where

r

is the radius of the circle.
Therefore,
Area of the circle with radius

r

π r^{2}

uni t^{2}

.
Area of the circle with radius

R

π R^{2}

uni t^{2}

.
Therefore,

Ratio of areas of given two circles = \frac{π r^{2}}{π R^{2}}

\Rightarrow Ratio of areas of given two circles = \frac{r^{2}}{R^{2}}

Substituting eq(1) in the above equation, we get,

Ratio of areas of given two circles = \frac{3^{2}}{4^{2}}

\Rightarrow Ratio of areas of given two circles = \frac{9}{16}

Therefore, if the circumferences of two circles are in the ratio

3 : 4

, then their areas are in the ratio

9 : 16

.
Hence, the correct option is 3.

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