Answer
$\frac{-\pi s+\sqrt{{{\left( \pi s \right)}^{2}}+4\pi A}}{2\pi }$
Work Step by Step
$A=\pi {{r}^{2}}+\pi rs$
Subtract $A$ on both sides of the equation.
$\begin{align}
& A=\pi {{r}^{2}}+\pi rs \\
& A-A=\pi {{r}^{2}}+\pi rs-A \\
& 0=\pi {{r}^{2}}+\pi rs-A
\end{align}$
Now, using the quadratic formula,
$r=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Substitute the values of the equation in the quadratic formula,
$a=\pi ,b=\pi s,c=-A$
Then, the equation is,
$\begin{align}
& r=\frac{-\pi s\pm \sqrt{{{\left( \pi s \right)}^{2}}-4\times \pi \times \left( -A \right)}}{2\times \pi } \\
& =\frac{-\pi s\pm \sqrt{{{\left( \pi s \right)}^{2}}+4\pi A}}{2\pi } \\
& =\frac{-\pi s\pm \sqrt{{{\left( \pi s \right)}^{2}}+4\pi A}}{2\pi }
\end{align}$
We only consider $ r \ge 0 $.
$r=\frac{-\pi s+\sqrt{{{\left( \pi s \right)}^{2}}+4\pi A}}{2\pi }$
Thus, the value is $r=\frac{-\pi s+\sqrt{{{\left( \pi s \right)}^{2}}+4\pi A}}{2\pi }$.
