Geodesic Ball on $\mathbb{R}^2$ with Riemannian Metrics that only depends on $r$

Question

I am having a problem that deals with the metric on $\mathbb{R}^2$ parametrized with polar coordinate $g=dr^2+f(r)^2d\theta^2$ for some $f$ such that $f(0)=0,f'(0)>0,f(r)>0$ for $r>0$. It asks me to show that the circle $S_\rho$,$r=\rho$ is the geodesic ball with radius $\rho$. It makes sense to me that a usual circle should be a geodesic ball since the metric is independent of $\theta$. But what makes me confused is about how to show that it is indeed with the same radius as the unit circle in tangent space. Is there any way to do this except for solving the geodesic equation explicitly?