When I cut the pieces for my octagons from a 4" strip, I cut the pieces that will square it off from a 2 3/4" strip. (This is based on Maxine Rosenthal's book, "One Block Wonders.")
While I don't know for sure, I suspect that if you subtract 1/2" (1/4" + 1/4") off each measurement for seam allowances, you'd come up with a repeatable proportion.
If so, then the ratio 3 1/2: 2 1/4, which can be reduced to 14:9, should work. For example, if you cut your octagon pieces from a 6" strip, then you should cut the squaring pieces from a 3.86" strip. In mathematical terms, 14/9:6/x (in English, 14 is to 9 as 6 is to x (the unknown width of the strip from which you'll cut squaring pieces) Working the math, 14x=54, therefore x=3.86").
That said, there are plenty of smarter quilters on this board than me! Since I've only been quilting for about 18 months, I may be all wrong in this approach to answering your question. I hope others will continue to chime in!