Overview

Let $r$ be the radius of the circle:

1. Relate area $A$ to radius $r$: $A = \pi r^2 \implies r^2 = \frac{A}{\pi} \implies r = \sqrt{\frac{A}{\pi}}$

2. Relate circumference $C$ to radius $r$: $C = 2\pi r$

3. Substitute $r = \sqrt{\frac{A}{\pi}}$ to express $C$ as a function of $A$: $C(A) = 2\pi\sqrt{\frac{A}{\pi}} = 2\sqrt{\frac{\pi^2 A}{\pi}} = 2\sqrt{\pi A}$