Question

The ratio of the circumference of two circles is 2:3. What is the ratio of their areas?
A. $ \dfrac{4}{9} $
B. $ \dfrac{2}{3} $
C. $ \dfrac{9}{4} $
D. $ \dfrac{3}{2} $

Answer 1

Hint: Since, radius is the common term in both Area & Circumference of a circle, so from the given ratio of the circumferences we need to find a ratio of the radius of two circles and then using the derived ratio we need to find the ratio of their areas.
Formula:
Circumference of a circle for a given radius (r) $ = 2\pi r $
Area of a circle for a given radius (r) $ = \pi {r^2} $

Complete step-by-step answer:
Let us consider the radius of the two circles to be $ {r_1} $ ​and $ {r_2} $ ​.
Then, according to the question, we have
Circumference of the first circle as $ 2\pi {r_1} $
Similarly, Circumference of the second circle as $ 2\pi {r_2} $
Hence, ratio of the circumferences of the two circles will be the ratio of $ 2\pi {r_1} $ & $ 2\pi {r_2} $ , i.e. $ \dfrac{{2\pi {r_{_1}}}}{{2\pi {r_2}}} = \dfrac{{{r_1}}}{{{r_2}}} $ .
But in the question, the ratio of circumferences of the two circles is given as 2:3, so $ \Rightarrow \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{2}{3} $
Now, the area of the first circle will be $ \pi r_1^2 $ and Area of the second circle will be $ \pi r_2^2 $
So, the ratio of the areas of the two circles will be \[\pi r_1^2/\pi r_2^2 \Rightarrow r_1^2/r_2^2 \Rightarrow {(\dfrac{{{r_1}}}{{{r_2}}})^2}\].
Now putting the value of $ \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{2}{3} $ in the derived ratio of the areas of the two circles we will be getting the value as
\[{(\dfrac{{{r_1}}}{{{r_2}}})^2} = {(\dfrac{2}{3})^2} = \dfrac{4}{9}\]
So, the ratio of the areas of two circles is $ \dfrac{4}{9} $
 So, the correct answer is “Option A”.

Note: Kindly remember the two formulas and just derive and conclude accordingly. And if possible remember the simple theory that squaring the ratio of the circumference will land us to the required ratio of the area of the same circles.