Question

If the area of the circle is $15400c{{m}^{2}}$ then its circumference is
\[\begin{align}
  & A.449cm \\
 & B.440cm \\
 & C.450cm \\
 & D.460cm \\
\end{align}\]

Answer 1

Hint: Here, approach is to know the concept of circle and most importantly the formula which we use for finding the area, $A=\pi {{r}^{2}}$ and circumference, $c=2\pi r$ of the circle. Here, find the radius of the circle from the given data i.e. area of circle and then use the value of radius in calculating the circumference of the circle. In this way, we will get the required answer.

Complete step-by-step answer:
Now, let's look over the given condition:
Here, we have an area of circle = $15400c{{m}^{2}}$.
And we have to find its circumference = ?
We know that there are two formulas that can be used to calculate the circumference of a circle, i.e. $c=2\pi r$.
Where, $\pi $ is the mathematical constant approximately equal to 3.14, r is equal to the radius and d is equal to the diameter.
Because the radius of a circle is equal to twice its diameter, these equations are essentially the same.

The units for circumference can be any unit for the measure of length, feet, miles, meters, centimeters etc.
But, here we have an area given in \[c{{m}^{2}}\] , so we have to find circumference in cm.
The area of a circle can be calculated using the diameter or the radius with two different formula:
$A=\pi {{r}^{2}}\Rightarrow A=\pi {{\left( \dfrac{d}{2} \right)}^{2}}$.
Since, $r=\dfrac{d}{2}$ these equations are essentially the same.
Now, area,
$\begin{align}
  & A=15400c{{m}^{2}} \\
 & \pi {{r}^{2}}=15400 \\
\end{align}$
Here, according to the option (i.e. no decimal point answers) we conclude that, we have to put a rational value of $\pi $.
i.e. $\pi =\dfrac{22}{7}\Rightarrow 3.14$
We are using this value in our question.
\[\begin{align}
  & \therefore \pi {{r}^{2}}=15400 \\
 & \Rightarrow \dfrac{22}{7}\times {{r}^{2}}=15400 \\
 & \Rightarrow {{r}^{2}}=\dfrac{15400\times 7}{22} \\
 & \Rightarrow {{r}^{2}}=4900 \\
 & \Rightarrow r=\sqrt{4900} \\
 & \Rightarrow r=70 \\
\end{align}\]
Now, we get r = 70 cm.
We know, formula for circumference i.e.
\[\begin{align}
  & c=2\pi r \\
 & \Rightarrow 2\times \dfrac{22}{7}\times 70 \\
 & \Rightarrow 44\times 10 \\
 & \Rightarrow 440cm \\
\end{align}\]
This is the required answer.

So, the correct answer is “Option B”.

Note: The use of value of $\pi $ is very crucial in calculating the area and circumference or wherever we find $\pi $ in equations. It is dependent on the students what he uses or he/she uses in the exam. Because there are choice based questions. So, we have to take care of options as well. (In which form it is given)
Generally, for removing confusion over the value of $\pi $ most of the entrance examination start giving approximate statement regarding use of value of constant (like $\pi $)
The $\pi $ is an irrational number represented by 3.14159265....