MCQ
The area of a circle whose area and circumference are numerically equal, is :
- A$2\pi\text{ sq. units}$
- ✓$4\pi\text{ sq. units}$
- C$6\pi\text{ sq. units}$
- D$8\pi\text{ sq. units}$
✓
Answer
Correct option: B.
$4\pi\text{ sq. units}$
We have given that circumference and area of a circle are numerically equal.
Let it be $x.$
Let $r$ be the radius of the circle,
therefore, circumference of the circle is $2\pi\text{r}$ and area of the circle will be $\pi\text{r}^2.$
Therefore, from the given condition we have,
$2\pi\text{ r}=\text{x}\ \dots(1)$
$\pi\text{ r}^2=\text{x}\ \dots(2)$
Therefore, from equation $(1)$ get $\text{r}=\frac{\text{x}}{2\pi}$
Now we will substitute this value in equation $(2)$
we get, $\pi\Big(\frac{\text{x}}{2\pi}\Big)^2=\text{x}$
Simplifying further we get,
$\pi\times\frac{\text{x}^2}{4\pi^2}=\text{x}$
Cancelling $x$ we get,
$\pi\times\frac{\text{x}}{4\pi^2}=1$
Now we will cancel $\pi$
$\frac{\text{x}}{4\pi}=1\ \dots(3)$
Now we will multiply both sides of the equation $(3)$ by $4\pi$ we get,
$\text{x}=4\pi$
Therefore, area of the circle is $4\pi\text{ sq. units}.$
Hence, option $(b)$ is correct.
Let it be $x.$
Let $r$ be the radius of the circle,
therefore, circumference of the circle is $2\pi\text{r}$ and area of the circle will be $\pi\text{r}^2.$
Therefore, from the given condition we have,
$2\pi\text{ r}=\text{x}\ \dots(1)$
$\pi\text{ r}^2=\text{x}\ \dots(2)$
Therefore, from equation $(1)$ get $\text{r}=\frac{\text{x}}{2\pi}$
Now we will substitute this value in equation $(2)$
we get, $\pi\Big(\frac{\text{x}}{2\pi}\Big)^2=\text{x}$
Simplifying further we get,
$\pi\times\frac{\text{x}^2}{4\pi^2}=\text{x}$
Cancelling $x$ we get,
$\pi\times\frac{\text{x}}{4\pi^2}=1$
Now we will cancel $\pi$
$\frac{\text{x}}{4\pi}=1\ \dots(3)$
Now we will multiply both sides of the equation $(3)$ by $4\pi$ we get,
$\text{x}=4\pi$
Therefore, area of the circle is $4\pi\text{ sq. units}.$
Hence, option $(b)$ is correct.
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