# If the circumference and the area of a circle

Question:

If the circumference and the area of a circle are numerically equal, then diameter of the circle is

(a) $\frac{\pi}{2}$

(b) $2 \pi$

(c) 2

(d) 4

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,

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

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

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

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

We can rewrite this equation as given below, $x=2 \times \pi \times 2$

Comparing equation (4) with equation (1) we get $r=2$

Therefore, radius of the circle is 2. We know that diameter of the circle is twice the radius of the circle.

$\therefore$ diameter $=2 \times$ radius

$\therefore$ diameter $=2 \times 2$

$\therefore$ diameter $=4$

Therefore, diameter of the circle is 4 .

Hence, option $(d)$ is correct.