What is the largest value of $r$ such that the following statement is always true?

"Let $C$ be a convex region with area $1$. There must exist a triangle contained in $C$ whose perimeter is at least $r$."

I don't need the actual largest value of $r$, but a lower bound would be nice. Using the fact that any convex region with unit area must contain a line segment of length $2/\sqrt{\pi}$, it is clear, for example, that $r\geq4/\sqrt{\pi}$ .