In this paper, we prove that for any bounded set of finite perimeter Ω ⊂ R, we can choose smooth sets E ⋐ Ω such that E → Ω in L and lim sup (Formula Presented) In the above Ω is the measure-theoretic interior of Ω, P(·) denotes the perimeter functional on sets, and C1(n) is a dimensional constant. Conversely, we prove that for any sets E ⋐ Ω satisfying E → Ω in L, there exists a dimensional constant C2(n) such that the following inequality holds: lim inf (Formula Presented) In particular, these results imply that for a bounded set Ω of finite perimeter,(Formula Presented) holds if and only if there exists a sequence of smooth sets E such that E ⋐ Ω, E → Ω in L and P(E) → P(Ω).