Slic3r estimates it automatically...

Well crap.  Haha.  I guess that is the easier way to go about it.

Here's what it looks like:

The volume of a cylinder, which is what the filament is, is:

V = pi * r^2 * h

Where r is roughly 0.875mm (depending on your filament's diameter/radius), so we can solve for h (which is the length of filament needed)

Also 0.875mm is equal to 0.0875cm

h = V/(pi * r^2)

h = V/(pi * (0.0875cm)^2)

h = V/(0.024cm^2)

Inverting that:

h = V * 41.667 cm^-2

This will vary with the radius of the filament, but it should give you a good estimate.

Note:  Make sure to get a good measurement, because the difference in filament diameter changes the number.  Filament with a 1.72mm diameter will give you a multiplier of 43 instead of 41.