*not*involving measuring tapes and plumb bobs, etc., but instead with scales during the weighing process. The math involved is at best junior high level. Please forgive me if some of this is elementary.

We’ll assume you know the primary datum and the arms for each of the wheels, where the plane is sitting on the scales. You start with the plane empty. Empty weight is, of course, the sum of the weights given by the three scales. The empty moment is the sum of the three moments, which is the weight on that scale multiplied by the corresponding arm. The empty CG is the empty weight divided by the empty moment.

Empty weight: EW = W1 + W2 + W3

Empty moment: EM = (W1 * arm1) + (W2 * arm2) + (W3 * arm3)

Empty CG = EM / EW

To determine the arm for any fuel, pilot/passenger, or baggage, weigh the airplane with that fuel, person, or baggage. Calculate the new moment as described above. Subtract the empty moment from the new moment to get the moment of whatever you added. Divide that by the weight of what you added, and you have the arm for that addition.

Loaded weight: LW = LW1 + LW2 + LW3

Added weight: AW = LW – EW (cross check this against the known added weight)

Loaded moment: LM = (LW1 * arm1) + (LW2 * arm2) + (LW3 * arm3)

Arm of added weight = (LM – EM) / AW

Loaded CG = LM / LW

Of course, for subsequent additions, base it on the last loaded weight, unless you removed that weight (back to empty) before adding the new load.