Question

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

Options

• \[\frac{\pi}{2}\]

•  2\[\frac{\pi}{2}\]

•  2

• 4

Answer 1

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 `2pir`and area of the circle will be `pir^2`.

Therefore, from the given condition we have, 

`2pir=x     .................(1)`

`pi r^2 = x ...............(2)`

Therefore, from equation (1) get `r=x/(2pi)`. Now we will substitute this value in equation 

we get, `pi(x/(2pi))^2=x`

Simplifying further we get, 

`pixx x^2/(4pi^2)=x`

Cancelling x we get,

`pi xx x/(4pi^2)=1`

Now we will cancel `pi` 

`x/(4pi)=1...........(3)`

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

We can rewrite this equation as given below, `x=2xxpixx2`

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.

`∴ "diameter"=2xx "radius"`

`∴ "diameter"=2xx2`

`∴"diameter"=4`

Therefore, diameter of the circle is 4.

Hence, option (d) is correct.