A geometry problem by Phil Barr

A geometry student decided to draw a circle using the length of a trash bag tie as the length of the radius. He made sure that the tie was straight and then he set his compass to the exact same length as the tie. After using the compass to draw the circle, he then decided to measure the circumference using the length of the tie as his unit of measure. He bent the tie so that its curvature would exactly match the curvature of the circle. Then he measured around the circumference, marking off arc lengths equal to the length of the tie. After marking off six of these arc lengths, he discovered that the remaining arc length was less than the length of the tie. What portion of the length of the tie was the remaining arc length?
(Give the answer as a decimal rounded to three places.)