Question

How do you write the area a of a circle as a function of its circumference?

Answer 1

Finding the relation between area and circumference by using the formula:

Let the area of circle be $A$ and circumference $C$.

Formula for the area of circle is given by (${\mathrm{\pi r}}^2$) where $r$ is the radius of the circle.

and the circumference is given by $2\mathrm{\pi r}$.

Where $\mathrm{\pi }$ is a constant and $r$ is the radius of the circle.

$\therefore C=2\left(\mathrm{\pi }\right)r$

$\Rightarrow$ $C^2=2^2{\left(\mathrm{\pi }\right)}^2{r}^2$ (Squaring both sides)

$\Rightarrow$ $C^2=4\left(\mathrm{\pi }\right)\left(\mathrm{\pi }\right)r^2$

$\Rightarrow$ $C^2=4\left(\mathrm{\pi }\right)\mathrm{A}$ $\left[\because A=\pi r^2\right]$

$\Rightarrow$ $A=\frac{C^2}{4\left(\mathrm{\pi }\right)}$

Therefore, area as a function of the circumference is given by $A=\frac{C^2}{4\left(\mathrm{\pi }\right)}$.