In each rotation which C makes, the radius of the arc it describes is 1 unit. In the first rotation, C turns through an angle of $120 °$, so it moves a distance $\frac{1}{3} \times 2 \times \pi \times 1$, that is $\frac{2\pi}{3}$.
As it is the centre of the second rotation, C does not move during it.
In the third rotation, C again turns through an angle of 120 °, so the total distance travelled is $2 \times \frac{2\pi}{3} = \frac{4\pi}{3}$.