Question

The ratio of circumference and area of a circle is 2 : 7. Find its circumference.
A. $14\pi $
B. $\dfrac{7}{\pi }$
C. $7\pi $
D. $\dfrac{14}{\pi }$

Answer 1

Hint: In order to solve this question, we will first consider the given ratio and will try to find the radius of the circle by using the formulas like, circumference of circle = $2\pi r$ and the area of circle = $\pi {{r}^{2}}$. And after getting the value of the radius, we can calculate the circumference to get the correct answer.

Complete step-by-step answer:
In this question, we have been asked to find the circumference of a circle whose circumference to the area ratio is 2 : 7. To solve this question, we will first consider the radius of the given circle as r. So, the circle would look like the figure below.

 We have been given that the ratio of the circumference and area of a circle is 2 : 7. So, we can say,
$\dfrac{Circumference\text{ }of\text{ }circle}{Area\text{ }of\text{ }circle}=\dfrac{2}{7}$
We know that the circumference and area of a circle are given by $2\pi r$ and $\pi {{r}^{2}}$ respectively. So, we can write the above equality as,
$\dfrac{2\pi r}{\pi {{r}^{2}}}=\dfrac{2}{7}$
We know that the common terms of the numerator and denominator can be cancelled out. So, we get,
$\begin{align}
  & \dfrac{2}{r}=\dfrac{2}{7} \\
 & \Rightarrow 2r=14 \\
 & \Rightarrow r=7 \\
\end{align}$
Hence, we can say that the radius of the circle is 7. Now, we have been asked to find the value of the circumference of the circle. So, we will put the value of radius in the formula of circumference of circle, that is, $2\pi r$. So, we get,
Circumference of the circle = $2\pi \times 7$. And so, the circumference of the circle = $14\pi $
Hence, we can say that the circumference of the circle is $14\pi $, where the ratio of the circumference to the area is 2 : 7.
Therefore, the correct answer is option A.

Note: There is a possibility of interchanging the formulas for circumference and area of circle, while solving this question. That would lead to an incorrect answer. Also we can make a mistake by assuming $\dfrac{Circumference\text{ }of\text{ }circle}{Area\text{ }of\text{ }circle}=\dfrac{2}{7}$ as $\dfrac{Area\text{ }of\text{ }circle}{Circumference\text{ }of\text{ }circle}=\dfrac{2}{7}$, which would also lead to an incorrect answer.