Answer
$6.928 \text{ cm}$
Work Step by Step
$\because A = \dfrac{1}{2}r^2 \theta$
where
$r$: Radius of the circle
$\theta$: Central angle that subtends the arc (in radians)
$A$: Area of the sector of the circle formed by the central angle $\theta$
Derive the formula for $r$:
\begin{align*}
A&=\frac{1}{2}r^2\theta\\\\
2A&=r^2\theta\\\\
\dfrac{2A}{\theta}&=r^2\\\\
\pm\sqrt{\dfrac{2A}{\theta}}&=r\\
\end{align*}
Since $r$ is a length, it has to be non-negative. So,
$r= \sqrt{\dfrac{2 A}{\theta}}$
Using this formula gives:
$r= \sqrt{\dfrac{2 A}{\theta}}$
$r = \sqrt{\dfrac{2 \times 6}{\frac{1}{4}}} =\sqrt{12 \cdot 4}= \sqrt{24} \approx \boxed{6.928 \text{ cm}}$
