Basically, all movements are (small) straight lines, the volume of a straight line is easily calculated as you already guessed.
To calculate the volume to be extruded you multiply the following parameters:
- the layer height (h)
- flow modifier (e.g. as pertectage) (SF)
- extruder nozzle diameter (dn)
- distance of the straight line (l)
With this volume you can calculate how much filament you need to extrude. To get the length (thus the length defined by the E parameter), divide the obtained volume by surface area of your used filament by:
- π * (filament radius)2 or alternatively π /4 * (filament diameter)2
To sum up, the value of E is given by:
$$ E_{value} = \frac{h \times {SF} \times d_{n} \times l}{\frac{\pi}{4}d_{f}^2} = \frac{4 \times h \times {SF} \times d_{n} \times l}{\pi \times d_{f}^2} $$