Question

The circumference of a circle is $30\pi $ . What is its area?
(a) $15\pi $
(b) $225\pi $
(c) $400\pi $
(d) $900\pi $

Answer 1

Hint: For solving this question firstly, we will find the value of the radius of the given circle by equating the value of given circumference with the standard result $2\pi r$ . After that, we will apply the formula $\pi {{r}^{2}}$ to calculate the value of the area of the given circle.

Complete step-by-step answer:
It is given that the value of circumference of the circle is $30\pi $ and we have to find the value of the area of the circle.
Now, before we proceed we should know the following two formulas:
1. For a circle of radius $r$ units, then the value of circumference is equal to $2\pi r$ units.
2. For a circle of radius $r$ units, then the value of the area of the circle is equal to $\pi {{r}^{2}}$ sq. units.
Now, to understand the physical meaning of circumference look at the figure given below:

In the above figure, the circumference of the circle will be the length of the periphery of the circle. For example: consider a cylindrical rod, and to measure the circumference of its cross-section, we can take the thread and wound it once, then the length of the thread wounded will be equal to the circumference of its cross-section.
Now, let the radius of the given circle is $r$ units and it is given that the value of its circumference is equal to $30\pi $ . Then,
$\begin{align}
  & 2\pi r=30\pi \\
 & \Rightarrow r=15 \\
\end{align}$
Now, from the above result, we can say that the value of radius will be equal to 15 units. And for the value of the area for the circle, we will use the formula given in the second point. Then,
$\begin{align}
  & Area=\pi {{r}^{2}} \\
 & \Rightarrow Area=\pi \times {{\left( 15 \right)}^{2}} \\
 & \Rightarrow Area=225\pi \\
\end{align}$
Now, from the above result we conclude that the area of the value of the area of the given circle will be equal to $225\pi $ sq. units.
Hence, option (b) will be the correct option.

Note: Here, the student should first understand what is asked in the problem. After that, we should first find the value of the radius of the given circle and then calculate for the value of the area of the given circle. Moreover, though the problem is very easy but the student should avoid calculation mistakes while solving to get the correct answer.