In 1999 Zoltán Füredi and John E. Wetzel, two covering problems meisters, found a triangle with a remarkable property.[1]
It can cover[2] each and every triangle of perimeter 2. It is the smallest region (not just the smallest triangle) that can do this and it is unique. I made a drawing of it using GeoGebra and fitted some sample triangles with perimeter 2 inside it
The length of is , , the length of is and the perimeter of is about .
[1] The smallest convex cover for triangles of perimeter two, Geometriae Dedicata, 2000
[2] To be precise, it can cover a congruent copy of any such triangle.