I remember seeing somewhere that for every connected compact set $\Omega$ in $\mathbb{R}^2$ with piecewise $C^1$ boundary we have $$A(\Omega_r)\leq A(\Omega)+L(\partial \Omega)r+ \pi r^2,$$ where $$\Omega_r=\{x\in \mathbb{R}^2: d(x,\Omega)\leq r\},$$ $A$ denotes the area and $L$ the length. I tried to find a reference for this inequality, but I only found Steiner's formula which states that equality holds when $\Omega$ is convex. Can someone please give me a reference for that inequality?