Why not displacement?

The path of $A$ is a cycloid, with equation $x \; = \; R(\theta - \sin\theta) \hspace{1cm} y \; = \; R(1 - \cos\theta)$ Thus we calculate that $\frac{ds}{d\theta} \; = \; \sqrt{x'(\theta)^2 + y'(\theta)^2} \; = \; \sqrt{2R^2(1 - \cos\theta)} \; = \; \sqrt{4R^2\sin^2\tfrac12\theta}$ and hence $\frac{ds}{d\theta} \; = \; 2R\big|\sin\tfrac12\theta\big|$ The distance $A$ travels in a single revolution is $2R\int_0^{2\pi}\sin\tfrac12\theta\,d\theta \; =\; \boxed{8R}$