In fact, the configuration maximizing the area is the one that is convex and inscribed in a circle: look here for a proof.

It remains to find the radius of the circle. Writing for the lengths, we must have (considering the sum of the angles the center views the segments). This can be solved numerically. I doubt there's an explicit expression for . Does anyone know?

Then the angle between the first and second segment is , and it's the same for the others.