Family of Triangles and their Circumradii

Geometry Level 5

A family of triangles all have the same area which is numerically the same as their perimeters. The greatest circumradius of any of them is exactly twice the least circumradius of any of them. What's the area of a triangle in this family of triangles (to 3 decimal places)?

Note: An equilateral triangle with an area of \(12 \sqrt{3}\) also has a perimeter of \(12 \sqrt{3}\), but this is the only such triangle with this area and perimeter, and hence only one circumradius.

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