I could calculate the distance for you, but I would rather give you a better way to locate the centers of the flanges.
If I were doing it, I would start with a 37 inch diameter circle (18.5" radius) centered at the center of the stop sign. I would then draw a radius of the circle which bisects one of the sides of the the octagon. From there I would measure an 11.25° angle to one side or the other, and draw that radius out to a diameter.
I would draw another diameter perpendicular to the first so I had four 90° angles. I would bisect two adjacent 90° angles (drawing the bisecting lines out to diameters), so that the circle would have been divided into 8 45° segments. After bisecting these angles one more time, I would have the centers for all 16 of the flanges.
(I hope they still teach how to do all this in Geometry!
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