Very stupid question about curvature of a circle

Question

It is a well-known fact that curvature of a circle with radius $R$ is equal to $\frac{1}{R}$. But if I write parameterization like $\gamma=(R\cos\frac{\tau}{R},R\sin\frac{\tau}{R})$ (because it must be a natural parametrization with $\vert \frac{d\gamma}{d\tau}\vert=1$) and then differentiate it twice I obtain $\frac{d^2\gamma}{d\tau^2}=(-\frac{1}{R}\cos\frac{\tau}{R},-\frac{1}{R}\sin\frac{\tau}{R})$ which gives $k=\vert \frac{d^2\gamma}{d\tau^2} \vert = \frac{\sqrt2}{R}$. What am I missing?..