To find the length of AX, we use the tangent of the angle \(\frac{1}{6} \pi\) (half of \(\frac{1}{3} \pi\)) in the right triangle formed by O, A, and X.

\(\tan \frac{1}{6} \pi = \frac{AX}{12}\)

\(\tan \frac{1}{6} \pi = \frac{\sqrt{3}}{3}\)

Thus, \(\frac{\sqrt{3}}{3} = \frac{AX}{12}\)

Solving for \(AX\), we get:

\(AX = 12 \times \frac{\sqrt{3}}{3} = 4\sqrt{3}\)