If D is the discriminant of (z-r1)(z-r2)(z-r3), then D = (r1-r2)^2 (r1-r3)^2 (r2-r3)^2, r1,r2,r3 complex. |D|=|r1-r2|^2 |r1-r3|^2 |r2-r3|^2 =(abc)^2 (a,b,c are the side-lengths of the triangle of roots) =(4 R A)^2 (R = circumradius of triangle of roots; A = area of triangle) = 16 R^2 A^2 Of course, this formula doesn't work when the triangle flattens out -- e.g., all roots are on the real axis -- because A->0 and R-> oo. Better off to use _curvature_ k=1/R instead of radius, so at least we have |D| = 16 A^2 / k^2 = 16 (A/k)^2 where A->0 and k->0 simultaneously in the 3-real-roots case.