# The area of a circle whose area and circumference

Question:

The area of a circle whose area and circumference are numerically equal, is

(a) $2 \pi$ sq. units

(b) $4 \pi$ sq. units

(c) $6 \pi$ sq. units

(d) $8 \pi$ sq. units

Solution:

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 and area of the circle will be.

Therefore, from the given condition we have,

$2 \pi r=x \ldots \ldots \ldots(1)$

$\pi r^{2}=x$.........(2)

Therefore, from equation (1) get $r=\frac{x}{2 \pi}$. Now we will substitute this value in equation (2) we get, $\pi\left(\frac{x}{2 \pi}\right)^{2}=x$

Simplifying further we get,

$\pi \times \frac{x^{2}}{4 \pi^{2}}=x$

Cancelling x we get,

$\pi \times \frac{x}{4 \pi^{2}}=1$

Now we will cancel $\pi$

$\frac{x}{4 \pi}=1$.........(3)

Now we will multiply both sides of the equation (3) by $4 \pi$ we get,

$x=4 \pi$

Therefore, area of the circle is $4 \pi s q$.units.

Hence, option (b) is correct.