9.cg-calculation. Center of Gravity Calculation
The center of gravity (CG) is the point at which an aircraft would balance if suspended. Its location is critical to flight safety because it directly affects stability, controllability, and performance. Pilots are required by 14 CFR 91.9 and 91.103 to operate the aircraft within the limits established by the manufacturer, and computing the CG before flight is the practical method of demonstrating compliance.
The Underlying Formula
CG location is determined by the principle of moments. A moment is the product of a weight and its arm — the horizontal distance from the reference datum to the weight's location.
- Moment (in-lb) = Weight (lb) × Arm (in)
- CG (in) = Total Moment ÷ Total Weight
The datum is an imaginary vertical plane chosen by the manufacturer (often the firewall, wing leading edge, or a point ahead of the nose) from which all arms are measured. Arms aft of the datum are positive; arms forward are negative.
Standard Computation Method
The computation method (sometimes called the longhand method) is the foundation every pilot must understand, even when using a loading graph or app.
- List every item of weight: empty aircraft, pilot, passengers, each baggage compartment, and usable fuel.
- Record each item's weight and its arm from the POH/AFM Section 6.
- Multiply weight × arm for each item to get its moment.
- Sum all weights to get total weight.
- Sum all moments to get total moment.
- Divide total moment by total weight to get the CG in inches aft of datum.
- Compare total weight against maximum gross weight, and compare CG against the forward and aft limits in the CG envelope for that weight.
Standard weights used when actual weights are unavailable:
- Aviation gasoline (avgas): 6 lb/gal
- Jet A: 6.7 lb/gal
- Oil: 7.5 lb/gal (1.875 lb/qt)
- Water: 8.35 lb/gal
Worked Example
Assume a Cessna 172 with the following data:
- Empty weight: 1,500 lb at arm 39.0 in → moment 58,500 in-lb
- Pilot + front passenger: 340 lb at arm 37.0 in → moment 12,580 in-lb
- Rear passengers: 170 lb at arm 73.0 in → moment 12,410 in-lb
- Baggage area 1: 40 lb at arm 95.0 in → moment 3,800 in-lb
- Fuel, 38 gal usable × 6 lb/gal = 228 lb at arm 48.0 in → moment 10,944 in-lb
Totals:
- Total weight = 1,500 + 340 + 170 + 40 + 228 = 2,278 lb
- Total moment = 58,500 + 12,580 + 12,410 + 3,800 + 10,944 = 98,234 in-lb
- CG = 98,234 ÷ 2,278 = 43.12 in aft of datum
The pilot then verifies: (1) total weight ≤ max gross (2,300 lb for a typical 172), and (2) the point (2,278 lb, 43.12 in) falls inside the published CG envelope.
Loading Graph and Index Methods
Many POHs simplify computation by publishing a loading graph that converts weight to moment/1000 (an index) for each station, eliminating arm multiplication. Sum the indexes, plot total weight versus total index on the center of gravity moment envelope, and confirm the point lies within limits. The result is mathematically identical to the longhand method — the index is just the moment divided by a reduction factor (usually 1,000) to keep numbers manageable.
Effect of Fuel Burn and Weight Shifts
CG is not static. As fuel burns, the CG moves toward the empty-weight CG; depending on the fuel station's arm, this can drive CG forward or aft. To recompute CG after a load change, two shortcut formulas are used:
- Weight added/removed: New CG = (Old Total Moment ± Added Moment) ÷ (Old Weight ± Added Weight)
- Weight shifted: ΔCG = (Weight Shifted × Distance Shifted) ÷ Total Weight
Always verify both takeoff CG and landing CG (most-burned condition) remain within limits.
Why It Matters
- Forward of forward limit: higher stall speed, longer takeoff roll, possible inability to flare.
- Aft of aft limit: reduced longitudinal stability, lighter elevator forces, and — most dangerously — possible inability to recover from a stall or spin.
A disciplined pilot computes weight and balance for every flight whose loading is not identical to a previously computed configuration. The arithmetic is simple; the consequences of skipping it are not.