Question

What is the diameter of a circle whose area is $16\pi $ ?

Answer 1

Hint: We know that for a circle of radius r, the area is equal to $\pi {{r}^{2}}$ . Using this formula, equate it with the given area, and find the value of ${{r}^{2}}$ . Hence, find the radius by taking the square root. We must also remember that diameter of a circle, $d=2r$ .

Complete step by step solution:
Let the radius of the circle be r, as we have shown in the figure below.

We know that the area of this circle would be,
$Area=\pi {{r}^{2}}...\left( i \right)$
But it is given in the question that the area of the circle is $16\pi $ .
So, using equation (i) and the information given in previous statement, we can say that
$\pi {{r}^{2}}=16\pi ...\left( ii \right)$
Now, cancelling the term $\pi $ from both sides of equation (ii), we get
${{r}^{2}}=16$
It is clear that to calculate radius r, we need to take square root on both sides. Hence,
$r=\sqrt{16}$
$\Rightarrow r=\pm 4$
Here, r is the radius of the circle. Clearly, we know that the radius of any circle can never be negative.
$\therefore r\ne -4$
Hence, we get $r=4$ .
Thus, the radius of the circle is 4 units.
Let the diameter of the circle be d.
We know that for a circle of radius r, the diameter is defined by the equation,
$d=2r...\left( iii \right)$
We know from the previous calculations that $r=4$ .
So, now using the value of r in equation (iii), we get
$d=2\times 4$
$\Rightarrow d=8$
Hence, the diameter of the circle is 8 units.

Note: We must always remember that the square root of 16 is both, +4 and -4, and we must not ignore any one of these altogether. We should also keep in mind that we need to find the diameter of the circle. So, we should not stop just after finding out the radius.