Weight and Balance, with the Math Behind It
6% of the exam 2 min
Weight and balance (6%) and mathematics (8%) together: the terms, the CG arithmetic, adverse loading and ballast, plus the geometry, ratios and conversions the General tests on their own.
The vocabulary
The datum is an imaginary vertical plane, chosen by the manufacturer and given in the aircraft specifications or TCDS, from which horizontal measurements are taken. An arm is the distance from the datum, positive aft and negative forward. A moment is weight times arm, in inch-pounds. Empty weight includes fixed ballast, hydraulic fluid and residual fuel and oil; basic empty weight also includes full engine oil, which is standard for most aircraft certified after 1978, while licensed empty weight includes only residual oil. Useful load is maximum gross weight minus empty weight. Tare is anything on the scales that is not the aircraft, such as chocks, and is subtracted.
Finding the CG
List each item's weight and arm, multiply to get moments, add the weights and the moments, and divide total moment by total weight. Averaging the arms ignores how much each item weighs and is the classic wrong answer. Removing an item subtracts both its weight and its moment; taking weight off behind the CG moves the CG forward. Percent of MAC is (CG − LEMAC) ÷ MAC × 100.
Adverse loading and ballast
An adverse-loading check deliberately tries to push the CG past a limit. For the forward check, all useful load forward of the forward limit is loaded and everything behind it is left empty, except fuel behind the limit, which is entered as minimum fuel: METO horsepower divided by 2, in pounds, for a piston engine. If the CG is out of limits, the FAA handbook's ballast formula is: ballast = aircraft weight × distance out of limits ÷ distance between the ballast location and the desired CG. Jack an aircraft for weighing per the manufacturer, raising wing jacks together with the safety collars run down.
The math the General tests by itself
Circle area is π × r², with the radius, not the diameter. Cylinder displacement is piston area times stroke. The hypotenuse of a right triangle is the square root of the sum of the squared sides. Compression ratio is the volume at bottom center divided by the volume at top center. Gear speed is inversely proportional to the number of teeth. One inch is exactly 25.4 mm. Subtracting a negative number is adding, and multiplication comes before addition and subtraction.
Where people lose points
✗ "The CG is the average of all the arms."
✓ It is total moment divided by total weight.
✗ "The datum is where the aircraft balances."
✓ That is the CG. The datum is the manufacturer's reference plane.
✗ "Chocks left on the scale don't matter."
✓ They are tare and must be subtracted.
✗ "Use the diameter in π × r²."
✓ Use the radius; the diameter gives an area four times too large.
✗ "A bigger driven gear turns faster."
✓ It turns slower, with more torque.
Numbers to memorize
| Moment | Weight × arm (in-lb) |
| Center of gravity | Total moment ÷ total weight |
| Percent of MAC | (CG − LEMAC) ÷ MAC × 100 |
| Minimum fuel (piston) | METO hp ÷ 2, in pounds |
| Ballast | Weight × distance out ÷ distance ballast-to-desired CG |
| Useful load | Max gross − empty weight |
| Circle area | π × r² |
| Compression ratio | Volume at BDC ÷ volume at TDC |
| Inch to millimeter | 1 in = 25.4 mm |
Test yourself
No answers here on purpose — retrieving them is the practice. Drill this domain if any of these stall you.
- Compute the loaded CG of an airplane from three items of weight and arm.
- Explain what is loaded and left empty in a forward adverse-loading check.
- Write the FAA handbook's ballast formula from memory.
- Find the displacement of a cylinder with a 5-inch bore and 4-inch stroke.
- Convert 50 mm to inches and 3/8 inch to a decimal.
Go deeper on these
- Aviation fuel identification, fueling safety and fire classes
- Aircraft bolts, nuts, rivets and safety wire for the A&P General