Question

What is the circumference of a circle with an area of $49\pi \text{ }inche{{s}^{2}}?$

Answer 1

Hint: We use the basic formulae for the calculation of area and circumference of a circle. The formula to find the area of a circle is given by $A=\pi {{r}^{2}},$ where r is the radius of the circle and the formula to find the circumference of the circle is $C=2\pi r,$ where r again stands for radius of the circle. Using the value of the given circle’s area, we find its radius. Using this radius in the circumference formula, we find its circumference.

Complete step by step solution:
In order to solve this question, let us take the radius of the circle to be r inches. We know the area of a circle is given by the formula,
$\Rightarrow A=\pi {{r}^{2}}$

Given in the question, the area of the circle is $49\pi \text{ }inche{{s}^{2}}.$ Substituting this in the above formula, we calculate for the radius r of the circle.
$\Rightarrow 49\pi =\pi {{r}^{2}}$
Dividing both sides by $\pi ,$
$\Rightarrow 49={{r}^{2}}$
We now take the square root on both sides,
$\Rightarrow \sqrt{49}=r$
We know the square root of 49 is $\pm 7.$
$\Rightarrow \pm 7=r$
But we need to note that radius is a measure of distance and distance is always a positive quantity. Therefore, we consider only the positive value which is $+7.$
$\Rightarrow r=7\text{ }inches$
The circumference of a circle can be calculated using the formula,
$\Rightarrow C=2\pi r$
Substituting the value of the radius of the circle which is 7 inches,
$\Rightarrow C=2\pi \times 7$
We know the value of $\pi =\dfrac{22}{7}=3.14.$ Taking the fraction equivalent of $\pi ,$
$\Rightarrow C=2\times \dfrac{22}{7}\times 7$
Cancelling out the 7 in the numerator and denominator, we are left with
$\Rightarrow C=2\times 22$
Multiplying the two, we get the circumference of the circle as,
$\Rightarrow C=44\text{ }inches$
Hence, the circumference of a circle with an area of $49\pi \text{ }inche{{s}^{2}}$ is $44\text{ }inches.$

Note: It is important to note the formulae for calculating the area and circumference of basic geometrical figures in order to solve such sums. Care must be taken while considering the units. Only the area will be represented in $inche{{s}^{2}}$ since the formula has a term ${{r}^{2}}$ where r is the radius in $inches.$ The circumference of the circle has the units $inches.$