How fast does the required length of the wire to make a full circle around this welding wire roll increase each time it goes around on the roll?

Assuming the roll is one wire thick so to remove repeating the same circle on a wide roll. Each circle around must overlap the previous wire.

So the first trip around the roll is the shortest and as it spins on the roll the long it takes to complete each trip. How fast does the trip length increase? Is it logarithmic increase? Exponential?