What formula does the calculator use?

For each axis it sums mass × coordinate and divides by the total mass: CG_x = Σ(m·x) ÷ Σm, and the same for y and z. With masses 10, 20 and 15 at x = 0, 5 and 2, CG_x = (0 + 100 + 30) ÷ 45 = 2.889.

How do you calculate weight and balance?

Multiply each weight by its arm (its distance from the datum) to get its moment, add up the weights and the moments, then divide total moment by total weight. For 1,650 lb at 39 in, 340 lb at 37 in, 50 lb at 95 in and 240 lb of fuel at 48 in: 93,200 lb-in ÷ 2,280 lb = 40.88 in aft of the datum. Compare that with the forward and aft limits for the weight.

Larger aircraft give the CG as a percentage of the mean aerodynamic chord: %MAC = (CG arm − LEMAC) ÷ MAC length × 100, where LEMAC is the arm of the leading edge of the MAC. A CG at 40.88 in with LEMAC 30 in and MAC 60 in is at 18.1% MAC.

Does it matter what units I use?

No, as long as you are consistent. Use kilograms and metres, pounds and inches, or any other pair; the CG comes out in the distance unit you entered and moments in mass × distance (for example lb-in or kg-mm). Many flight manuals divide moments by 1,000 to keep the numbers short.

Can I use negative coordinates or arms?

Yes. Coordinates can be negative, zero or positive, which is useful when the origin or datum is inside the object. In weight and balance, items forward of the datum have negative arms and negative moments.

How does fuel burn move the CG?

Burned fuel removes its weight and its moment (weight × fuel arm). If the fuel arm is aft of the takeoff CG, the CG moves forward as fuel burns, and vice versa. The landing line shows the new weight and CG so you can check both ends of the flight.

How many components can I enter?

Up to 20 point masses and up to 12 weight and balance items. For larger assemblies, combine sub-assemblies into one row using their own mass and CG.

Center of Gravity Calculator

CG = Σ(m·x) ÷ Σm on each axis. Enter masses and coordinates for a 1D, 2D or 3D center of gravity, or weights and arms for a weight and balance: moment = weight × arm, CG = total moment ÷ total weight.

Mass Components

Mode:

Center of Gravity

Enter masses and coordinates, then click Calculate CG.

Formula Used

ΣM_x = Σ(m·x), CG_x = ΣM_x / Σm

ΣM_y = Σ(m·y), CG_y = ΣM_y / Σm

All axes are computed independently as a mass-weighted average.

Weight and balance (moment and CG arm)

Moment = weight × arm, CG arm = total moment ÷ total weight. Arms are measured from the datum; use negative arms forward of it. The default numbers are a generic example, not data for a real aircraft.

Units

ItemWeight (lb)Arm (in)Moment

— —

Fuel in US gal × 6.0 lb/gal

Forward CG limit

Aft CG limit

Max weight

Fuel burn

LEMAC arm

MAC length

Takeoff / loaded

Total weight: —
Total moment: —
CG arm: —
%MAC: —

Landing (after fuel burn)

Enter a fuel burn to see the landing weight and CG.

Worked examples

Three point masses (2D). 10 kg at (0, 0), 20 kg at (5, 3) and 15 kg at (2, 7). Σm = 45, ΣM_x = 0 + 100 + 30 = 130, ΣM_y = 0 + 60 + 105 = 165. CG = (130 ÷ 45, 165 ÷ 45) = (2.889, 3.667).
Beam with two loads (1D). 200 N at 1 m and 600 N at 4 m from the left end: CG = (200 × 1 + 600 × 4) ÷ 800 = 3.25 m. A single support there balances the beam.
Weight and balance. Empty 1,650 lb at 39.0 in (64,350 lb-in), front seats 340 lb at 37.0 in (12,580), baggage 50 lb at 95.0 in (4,750), 40 gal of avgas = 240 lb at 48.0 in (11,520). Total 2,280 lb and 93,200 lb-in, so CG = 40.88 in, inside 35.0–47.3 in and 120 lb under a 2,400 lb maximum. Burning 30 gal (180 lb, 8,640 lb-in) gives 2,100 lb at 40.27 in.
%MAC. With LEMAC at 30 in and a 60 in MAC, a CG of 40.88 in is (40.88 − 30) ÷ 60 × 100 = 18.1% MAC.

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Summary

The Center of Gravity Calculator finds the mass-weighted average position of up to 20 point masses along one, two or three axes and plots them with the combined CG. A second panel does a weight and balance the way pilots, loadmasters and vehicle builders do it: each item's weight times its arm from a datum gives a moment, and the total moment divided by the total weight gives the CG arm. Add forward and aft CG limits, a maximum weight, fuel in gallons or litres with a planned burn, and the leading edge and length of the mean aerodynamic chord to see whether the load is within limits at takeoff and landing and where the CG sits in %MAC. Use the manufacturer's data for any real aircraft.

How it works

Point masses: choose 1D, 2D or 3D and enter each mass with its coordinates from one reference origin. Any consistent units work.
The calculator sums the first moments ΣM = Σ(m·r) on each axis and divides by the total mass: CG = ΣM ÷ Σm. The plot shows each mass (circle size grows with mass) and the CG cross.
Weight and balance: enter each item's weight and arm (distance from the datum, negative if forward of it). Moment = weight × arm, and the CG arm = total moment ÷ total weight.
Fuel is entered as volume and converted with a standard density: avgas 6.0 lb/US gal (0.72 kg/L), Jet A 6.7 lb/US gal (0.80 kg/L). A planned burn gives the landing weight and CG.
Optional forward and aft CG limits and a maximum weight show whether the load is within the envelope. With LEMAC and MAC length the CG is also given as %MAC = (CG − LEMAC) ÷ MAC × 100.

Use cases

Find the CG of an assembly, vehicle chassis, robot or product from its components.
Run a weight and balance for a light aircraft, drone or RC model before flight.
Check a trailer, truck bed or cargo pallet load against a position limit.
Compute the centroid of a composite shape by entering each part's area as the "mass".
See how moving one heavy item changes the combined CG.
Teach first moments and weighted averages in statics or physics.

Frequently Asked Questions

Last updated: 2026-10-10 · Reviewed by Nham Vu