I think one way of expressing the hard problem is "given a set of edge-lengths {Ei}, determine the radius of the circumscribing circle".

IIRC, the formula for the circumradius of a triangle contains Hero's area formula as a term.

just by chance, i was working on something related to this the other day. the formula for the circumradius of a convex cyclic quadrilateral, with side lengths a , b , c and d is: /------------------------------------------------------------------ / (ab + cd) (ac + bd) (ad + bc) R = \ / ------------------------------------------------------------------ \/ (-a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d) the formula for the circumradius of a triangle can be obtained by putting d = 0 into this formula. mike