To find the length of the arc formed by a central angle in a circle, you can use the formula for arc length:
\[ L = r \cdot \theta \]
where:
- \(L\) is the arc length,
- \(r\) is the radius, and
- \(\theta\) is the central angle in radians.
In this case:
- The radius \(r = 27\) mm,
- The central angle \(\theta = \frac{2\pi}{3}\) radians.
Substituting the values into the formula:
\[ L = 27 \cdot \frac{2\pi}{3} \]
Calculating this gives:
\[ L = 27 \cdot \frac{2\pi}{3} = \frac{54\pi}{3} = 18\pi , \text{mm} \]
Therefore, the length of the arc is:
\[ \boxed{18\pi} , \text{mm} \]
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