Formula for coiling wire?

There are several formulae of differing complexities. Google, you should be able to find an online calculator.

Simplest method is calculating each wrap length is the circumference of the circle, πd for each wrap. Diameter d = drum outside diameter + thickness of cable. Number of wraps per layer is inside width of reel divided by cable width. 2nd layer, if any, is same, but has diameter of underlying layer + 2 times cable thickness.

Good luck!

Edit:

I thought you were winding cable on a reel, but it looks like you want to wind a coil. So, here's your example:

wire dia: 2.769 mm

Core dia: 20 mm

Wire Length: =20ft * 0.3048m/ft = 6.096 m

Circ: = π(2.769+20) = 71.53092313 mm

wraps: 6096mm / 71.53092313mm = 85.22188353

Coil Length: 85.22188353 wraps * 2.769 mm/wrap = 235.9793955 mm