Aloha :)
$$L=\int\limits_0^{2\pi}du(t)=\int\limits_0^{2\pi}\left\|\frac{d\vec u}{dt}\right\|\,dt=\int\limits_0^{2\pi}\left\|\binom{r-r\cos t}{r\sin t}\right\|\,dt=r\int\limits_0^{2\pi}\sqrt{(1-\cos t)^2+\sin^2t}\,dt$$$$\phantom{L}=r\int\limits_0^{2\pi}\sqrt{1-2\cos t+\cos^2t+\sin^2t}\,dt=r\int\limits_0^{2\pi}\sqrt{2-2\cos t}\,dt=r\int\limits_0^{2\pi}\sqrt{4\sin^2\frac{t}{2}}\,dt$$$$\phantom{L}=r\int\limits_0^{2\pi}2\sin\frac{t}{2}\,dt=r\left[-4\cos\frac{t}{2}\right]_0^{2\pi}=-4r\left(\cos\pi-\cos0\right)=-4r(-1-1)=8r$$
