To find the radius of the circle when given the arc length and the central angle, we can use the formula for the arc length of a sector:
\[ L = r \theta \]
where \(L\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.
First, we need to convert the given central angle from degrees to radians. The conversion from degrees to radians is done using the formula:
\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]
For \(\theta = 210\) degrees, we have:
\[ \theta = 210 \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6} \text{ radians} \]
Now we know that the arc length \(L\) is given as \(\pi\) units. Substituting the known values into the arc length formula:
\[ \pi = r \cdot \frac{7\pi}{6} \]
To solve for \(r\), we can rearrange the equation:
\[ r = \frac{\pi}{\frac{7\pi}{6}} = \pi \cdot \frac{6}{7\pi} = \frac{6}{7} \]
Thus, the radius \(r\) of the circle is:
\[ \boxed{\frac{6}{7}} \] units.