One possible formulation is to consider a smooth closed curve $\Gamma$ and to define the track as the Minkowski sum of $\Gamma$ and the circle of radius $r$. If $\Gamma$ is nice enough, so that the curvature at any point doesn't exceed $1/r$, then the outer boundary has length equal to the length of the inner boundary plus $2\pi 4\pi r$. (That is, the perimeter of the Minkowski sum of two regions is equal to the sum of the individual perimeters in most nice cases.)
One possible formulation is to consider a smooth closed curve $\Gamma$ and to define the track as the Minkowski sum of $\Gamma$ and the circle of radius $r$. If $\Gamma$ is nice enough, so that the curvature at any point doesn't exceed $1/r$, then the outer boundary has length equal to the length of the inner boundary plus $2\pi r$. (That is, the perimeter of the Minkowski sum of two regions is equal to the sum of the individual perimeters in most nice cases.)