Cv is proportional to the open area and a disch coeff.
the problem with ball valves is that they are classified as high recovery valves. Check Appendix A, Crane Technical Paper 410.
There are formulas relating Cv and Equivalents of Resistance Coefficient (i.e. Loss Coefficient) K for geometry of flow through a circular hole.
I call the above equation, eq.(2), i.e. Cv/area = f(K)
You started with a known Cv.  I call this equation, eq.(1).
Equation (1) divided by equation (2) gives you the effective flow area, provided you know how to calculate the Loss Coefficient,K, of your flow area. [a triangular shape]
For V-port, I would consider it as a sudden expansion and therefore K inlet = 0.5, K outlet = 1.0, and therefore K total = 1.5 (maximum). 
Hence Cv / (Cv/area) = [Cv] known / [38/sq.root(K)] = area, were K = 1.5 in this case. [My best approximation].
The area calculated is the effective flow area because it takes the loss coefficient of the geometry (obstacle or channel) into account.
Ref.1: Granger, Fluid Mechanics. [K for needle valves & gate valves]
Ref. 2: White, Fluid Mehanics. [K for butterfly valves].

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