Question

If the circumference of a circle is equal to the perimeter of a square what is the ratio of the area of the circle to the area of the square?
\[
  A\,\,\,\,22:7 \\
  B\,\,\,\,14:11 \\
  C\,\,\,\,11:7 \\
  D\,\,\,\,4:1 \\
 \]

Answer 1

Hint: Here we use the condition given in the question that circumference of a circle is equal to the perimeter of the square. We use the formulae area of circle $\pi {r^2}$ and area of square ${a^2}$.

Complete Step by Step Solution:
The objective of the problem is to find the ratio of the area of the circle to the area of the square.
Given that the circumference of the circle is equal to the perimeter of a square.
That is circumference of circle=perimeter of square.
The circumference of the circle is $2\pi r$ .
The perimeter of the square is $4a$ .
Now we get , $2\pi r = 4a$
Dividing with $2\pi $ on both sides, we get
$
   \Rightarrow \dfrac{{2\pi r}}{{2\pi }} = \dfrac{{4a}}{{2\pi }} \\
   \Rightarrow r = \dfrac{{2a}}{\pi } \\
 $
Dividing with a on both sides we get
$
   \Rightarrow \dfrac{r}{a} = \dfrac{{2a}}{\pi } \times \dfrac{1}{a} \\
   \Rightarrow \dfrac{r}{a} = \dfrac{2}{\pi } \\
 $
Now we have to find ratio of the area of the circle to the area of the square
That is we can write it as $\dfrac{{area\,of\,circle}}{{area\,of\,square}}$
We know that area of circle is $\pi {r^2}$ and area of square is ${a^2}$
Now , $\dfrac{{area\,of\,circle}}{{area\,of\,square}} = \dfrac{{\pi {r^2}}}{{{a^2}}}$
We can write this as $\pi {\left( {\dfrac{r}{a}} \right)^2}$
Now substitute $\dfrac{r}{a} = \dfrac{2}{\pi }$ , we get
$
   \Rightarrow \pi {\left( {\dfrac{2}{\pi }} \right)^2} \\
   \Rightarrow \pi \left( {\dfrac{{{2^2}}}{{{\pi ^2}}}} \right) \\
   \Rightarrow \dfrac{4}{\pi } \\
 $
Now taking $\pi $ as $\dfrac{{22}}{7}$. On substituting we get
$
   \Rightarrow \dfrac{4}{{\dfrac{{22}}{7}}} \\
   \Rightarrow \dfrac{{4 \times 7}}{{22}} \\
   \Rightarrow \dfrac{{28}}{{22}} \\
 $
On cancellation with 2 we get $\dfrac{{14}}{{11}}$
Therefore the ratio of the area of the circle to the area of the square is $14:11$.

Thus option B) is the correct answer.

Note:
The circumference of the circle is equal to $2\pi r$ and the perimeter of the square is equal to the $4a$. Remember that the words circumference and perimeter means the same. Perimeter or circumference is defined as the boundary that surrounds any shape or the length of the out line that surrounds any shape.