Answer
$r=\pm\dfrac{\sqrt{2A\theta}}{\theta}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of equality to solve the given equation, $
A=\dfrac{1}{2}r^2\theta
,$ for $r.$
$\bf{\text{Solution Details:}}$
Multiplying both sides by $2$ and then dividing by $\theta,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
2A=r^2\theta
\\\\
\dfrac{2A}{\theta}=r^2
\\\\
r^2=\dfrac{2A}{\theta}
.\end{array}
Taking the square root of both sides (Square Root Principle), the equation above is equivalent to
\begin{array}{l}\require{cancel}
r=\pm\sqrt{\dfrac{2A}{\theta}}
.\end{array}
Multiplying the radicand by an expression equal to $1$ so that the denominator becomes a perfect root of the index results to
\begin{array}{l}\require{cancel}
r=\pm\sqrt{\dfrac{2A}{\theta}\cdot\dfrac{\theta}{\theta}}
\\\\
r=\pm\sqrt{\dfrac{2A\theta}{\theta^2}}
\\\\
r=\pm\sqrt{\dfrac{1}{\theta^2}\cdot2A\theta}
\\\\
r=\pm\sqrt{\left(\dfrac{1}{\theta}\right)^2\cdot2A\theta}
\\\\
r=\pm\dfrac{1}{\theta}\sqrt{2A\theta}
\\\\
r=\pm\dfrac{\sqrt{2A\theta}}{\theta}
.\end{array}
