Answer
$r=\dfrac{\sqrt{A\pi}}{2\pi}$
Work Step by Step
Multiplying the denominator by an expression that will make it a perfect root of the index, then the rationalized-denominator form of the given equation, $
r=\sqrt{\dfrac{A}{4\pi}}
,$ is
\begin{array}{l}\require{cancel}
r=\sqrt{\dfrac{A}{4\pi}\cdot\dfrac{\pi}{\pi}}
\\\\
r=\sqrt{\dfrac{A\pi}{4\pi^2}}
\\\\
r=\sqrt{\dfrac{A\pi}{(2\pi)^2}}
\\\\
r=\dfrac{\sqrt{A\pi}}{2\pi}
.\end{array}
