To find the length of the arc formed by a central angle in a circle, you can use the formula for the arc length \( L \):
\[ L = r \theta \]
where:
- \( L \) is the arc length,
- \( r \) is the radius,
- \( \theta \) is the central angle in radians.
Given:
- \( r = 27 \) mm,
- \( \theta = \frac{2\pi}{3} \) radians.
Now, substitute the values into the formula:
\[ L = 27 \cdot \frac{2\pi}{3} \]
Now, perform the multiplication:
\[ L = \frac{27 \cdot 2\pi}{3} \]
Simplify:
\[ L = \frac{54\pi}{3} = 18\pi , \text{mm} \]
Thus, the length of the arc is
\[ \boxed{18\pi , \text{mm}} \]