Radius of super O track

I couldn't agree with you more, Steve.  Here is a recent post of mine on this topic:

I am convinced that the original scheme for O27 and (so-called) O31 tubular track was to coordinate the curved and straight sections so that the joints in a siding alongside a straight main line would match those of the main line.  This assures that a passing siding can be made using only standard sections.  With 8 curved sections in a circle, this scheme effectively requires that the radius be the length of a straight section multiplied by the square-root of 2.

It's pretty clear that Lionel intended their straight section to be exactly 10 inches, with the result that the radius is 14.142136 inches, the value that Rob and I agree on.  (I think we're using the same reasoning.)  When you double this to get the diameter, and add a tie length of 2.25 inches, the overall diameter comes to 30.534271, which is where we get the modern nominal diameter of O31 (but occasionally O30 for the same track).

O27, which I believe Lionel got from Ives, is the other way around.  It seems to have been designed with a round number for the overall diameter.  So, subtracting the somewhat shorter 2-inch tie and dividing by 2 gives a radius of 12.5 inches.  Then, dividing by the square-root of 2, we can back into the "correct" length for the straight section, which comes to an awkward 8.838835 inches.  I think that track manufacturers have not generally understood where this number may come from, because straight O27 sections vary quite a bit around this length.  Marx was particularly inconsistent.

As Fred suggested, it is quite tedious to get a good measurement, particularly by the obvious method of putting together a full circle and measuring it directly.  As Rob pointed out, the joints can vary in tightness; and the track itself is quite flexible (one of its virtues, actually).  It is essential to make at least two measurements, at right angles, to have any hope of accuracy.

Because of the difficulty of making a full-circle measurement, even if you are lucky enough to have enough sections for that, I devised the following method that requires only a single section and knowledge of how many sections a circle comprises.  It has the useful feature of working better, the more sections are needed for a circle.  It is also fairly insensitive to whether the section measured may have been bent a little straighter or more curved than it should be.  I have posted it several times before; here it is again: 

Measure the chord of a curved section, in a straight line from one end to the other of the center rail.  Multiply that number by half the cosecant of half the angle that the section turns.  The result is the radius to the center rail.

For example, O42 has 12 sections in a complete circle; so each section turns through 30 degrees.  Therefore you multiply by half the cosecant of 15 degrees, or 1.931852.  (For 8 sections in a circle, multiply by 1.306563; and for 16 sections, by 2.562915.)

The more careful you are measuring to the exact center of the end of the center rail, the more accurate your result will be.