What Is a Righting Arm (GZ) Calculator?
The righting arm, GZ, is the horizontal lever between a heeled vessel's center of gravity and its center of buoyancy — the force that pushes it back upright. This calculator uses the standard small-angle approximation, GZ = GM × sin(heel angle), to estimate that lever and the resulting righting moment from your vessel's metacentric height, heel angle and displacement.
It's the natural next step after calculating metacentric height (GM): GM tells you the initial stability tendency, while GZ at a specific heel angle tells you the actual size of the righting force at that moment.
How to Read Your Results
Righting Arm (GZ)
The estimated lever arm, in feet, pushing the vessel back toward upright at the entered heel angle. Larger values mean a stronger righting tendency at that angle.
Righting Moment
GZ multiplied by displacement, giving the actual righting moment in foot-tons — the turning force available to right the vessel at that heel angle and loading condition.
Heel Angle Used
The heel angle you entered, echoed back for confirmation.
Formula Validity
A flag noting whether your entered heel angle falls within the roughly 10-15° range where the small-angle approximation stays accurate, or whether you've gone beyond it and should refer to a full GZ stability curve instead.
Real-World Example
A vessel with GM of 2.0 ft, displacement of 150 long tons, heeled to 10 degrees:
| Item | Value |
|---|---|
| GM | 2.0 ft |
| Heel angle | 10° |
| sin(10°) | 0.1736 |
| Righting arm (GZ) | ≈0.347 ft |
| Righting moment | ≈52.1 ft-tons |
At the same GM but heeled to 25 degrees, the small-angle formula would give GZ ≈ 0.845 ft — but at that larger angle the approximation is already stretched past its reliable 10-15° range, so a real assessment would need the vessel's actual GZ curve rather than this linear estimate.
Tips for Using Righting Arm Estimates
- Stay within small angles. This formula is a starting approximation, not a substitute for a full GZ curve at larger heel angles.
- Use matching GM and displacement values from the same loading condition for a consistent result.
- Watch the trend, not just one number. Compare GZ at a few different heel angles to see how quickly righting force builds or falls off.
- Never use this as a sole safety determination — refer to your vessel's official stability booklet for regulated limits.
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Last updated: August 2026 · Reviewed by: Simple Calculator Tools Editorial Team