I guess from what I see, that [itex]d[/itex] is the diameter of the disc and [itex]\phi[/itex] is the angle the disc rotated. Then the radius of the disc would be [itex]d/2[/itex]. So the length of the string unwound will be radius times the angle rotated, or [itex]\frac{d}{2} \phi[/itex].