jmc

We begin by considering a point $P$ which is where the circle first touches a line $L.$



If a circle makes one complete revolution, the point $P$ moves to $P^{\prime }$ and the distance $P{P}^{\prime }$ is the circumference of the circle, or $2\pi \text{ m}.$

If we now complete the rectangle, we can see that the distance the center travels is $C{C}^{\prime }$ which is exactly equal to $P{P}^{\prime }$ or $\boxed{2\pi }$ meters.