Nice topic. This method was how I made my models here, except that I did the trigonometry on paper+calculator and/or in Microsoft Excel. (Actually I've never tried the hinge align tool!) Given any right triangle, it was easy to find a triangle with all sides ~= integers by copy/pasting equations until they would yield a size that fit (<0.05 stud error).
In cases where I couldn't calculate the angle beforehand, I would rotate the side of a polyhedron until it would collide in either direction, then average the two collision angles, manually input the predicted angle (trial and error), then take the average of the two most extreme angles that LDD would accept. A completed polyhedron with no errors (doublecheck by saving, closing LDD, and reopening the file) was always a good reason to celebrate.
At other times, LDD errors could be overcome (manipulated) by adding connected hinges+plates of polygons in different sequences. The update from LDD3 to LDD4 solved a lot of the leeway though; LDD is more accurate now than it had been.