Question

If the area and the circumference of the circle are numerically equal, then the radius of the circle is _______
\[\left( a \right)\dfrac{5}{2}\]
\[\left( b \right)2\]
\[\left( c \right)1\]
\[\left( d \right)\dfrac{2}{5}\]

Answer 1

Hint: We are asked to find the radius of a circle whose circumference and the area are the same. We will assume our required radius as r. Now, we will find the area using the formula, \[\text{Area}=\pi \times {{\left( \text{radius} \right)}^{2}}\] and circumference using \[\text{Circumference}=2\times \pi \times \text{ radius}\] and then equate them as they are equal. So, we equate them and solve for the radius to get our answer.

Complete step-by-step answer:
We have to find the volume of the radius of a circle whose circumference and area are numerically equal. Let us assume that r be the required radius.

We know that the circumference is defined as the total length of the circle and is given as
\[\text{Circumference}=2\times \pi \times \text{ radius}\]
As we have a radius as r, so, we get,
\[\text{Circumference}=2\pi r......\left( i \right)\]
Now, we know that the area of the circle is defined as the region occupied inside the circular shape. The area of the circle is given by the formula as,
\[\text{Area}=\pi \times {{\left( \text{radius} \right)}^{2}}\]
As the radius is r, so,
\[\text{Area}=\pi {{r}^{2}}......\left( ii \right)\]
Now, we have that area is numerically equal to the circumference.
Area = Circumference
Now using (i) and (ii), we get,
\[\Rightarrow 2\pi r=\pi {{r}^{2}}\]
Cancelling the like term, we get,
\[\Rightarrow 2r={{r}^{2}}\]
Simplifying, we get,
\[\Rightarrow 2r-{{r}^{2}}=0\]
Taking r common, we get,
\[\Rightarrow r\left( 2-r \right)=0\]
So, with r = 0 or 2 – r = 0, either r = 0 or r = 2.
As r = 0 is not in the option, so hence r = 2 is the answer.

So, the correct answer is “Option b”.

Note: One could solve this question like,
\[2r={{r}^{2}}\]
Cancelling r on both the sides, we get,
\[\Rightarrow 2=r\]
Therefore, we get the radius as 2.
But whenever we are not mentioned that r is not equal to 0, so we cannot divide the equation by that. That is,
\[2r={{r}^{2}}\]
Dividing both the sides by r, we get,
\[\Rightarrow \dfrac{2r}{r}=\dfrac{{{r}^{2}}}{r}\]
If r = 0, then this division is not valid. So we need to solve as mentioned in the solution.