To find the length of the arc formed by a central angle in a circle, we can use the formula:
\[ \text{Arc Length} = \theta \times r \]
where \(\theta\) is the central angle in radians and \(r\) is the radius of the circle.
First, we need to convert the central angle from degrees to radians. The conversion from degrees to radians is given by:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Given that the central angle is \(125\) degrees, we can convert it:
\[ \theta = 125 \times \frac{\pi}{180} = \frac{125\pi}{180} = \frac{25\pi}{36} \text{ radians} \]
Now, we can substitute \(\theta\) and the radius \(r = 27\) mm into the arc length formula:
\[ \text{Arc Length} = \frac{25\pi}{36} \times 27 \]
Now, simplifying this:
\[ \text{Arc Length} = \frac{25 \times 27\pi}{36} \]
Performing the multiplication:
\[ 25 \times 27 = 675 \]
Now we substitute back:
\[ \text{Arc Length} = \frac{675\pi}{36} \]
We can simplify this fraction by dividing both the numerator and the denominator by 9:
\[ \frac{675 \div 9}{36 \div 9} = \frac{75}{4} \]
Thus, the length of the arc is:
\[ \text{Arc Length} = \frac{75\pi}{4} \text{ millimeters} \]
So the final answer is:
\[ \frac{75\pi}{4} \text{ mm} \]