Why this conversion matters for rotary and turned work
On a 4th axis rotary table or a lathe C-axis, features like engraving, drilling, or milling around the outside of a cylindrical part are programmed in degrees of rotation, but the actual surface distance the tool travels depends on the part diameter. A small-diameter part covers far less surface distance per degree than a large one, which matters when working out engraving spacing or checking a feed rate against surface speed.
How the numbers are calculated
Arc length at the surface equals pi multiplied by the part diameter, multiplied by the rotary angle divided by 360, which is the same circumference-times-fraction relationship used for any arc on a circle. Fraction of a full revolution is simply the rotary angle divided by 360, showing what portion of one complete rotation the move represents.
Practical tips
Use the surface arc length to check that engraved text or repeated features spaced evenly around a part will actually fit without overlapping, since spacing that looks fine in degrees can crowd together on a small-diameter part. This calculation assumes the tool is cutting exactly at the part's outer diameter, so adjust the diameter input if the actual feature sits at a different radius, such as a groove or step partway down the part.