Compute heel angle, righting arm, or metacentric height. Check units, examples, and plotted stability relationships. Build fast engineering estimates with clean tables and downloads.
This table shows sample GM and GZ pairs with derived heel angles.
| Case | GM (m) | GZ (m) | Angle (deg) | Comment |
|---|---|---|---|---|
| Case A | 0.45 | 0.06 | 7.66 | Suitable small-angle check |
| Case B | 0.60 | 0.10 | 9.59 | Suitable small-angle check |
| Case C | 0.80 | 0.14 | 10.08 | Suitable small-angle check |
| Case D | 1.00 | 0.18 | 10.37 | Suitable small-angle check |
| Case E | 1.20 | 0.22 | 10.56 | Suitable small-angle check |
The calculator uses the small-angle initial stability relation.
Righting arm: GZ = GM × sin(theta)
Heel angle: theta = asin(GZ / GM)
Metacentric height: GM = GZ / sin(theta)
Approximate righting moment: Moment = Weight × GZ
Use consistent units for GM and GZ. The angle must be in degrees here. This simplified relation is strongest at small heel angles. Large-angle checks need full hydrostatic or cross-curve data.
The G-M angle relation helps engineers check early vessel stability. GM is the metacentric height. It reflects the distance between the center of gravity and the metacenter. GZ is the righting arm. It represents the lever that creates a restoring effect during heel. When the angle is small, GZ is commonly estimated from GM and the sine of heel angle. This makes the method fast and useful during concept design, training, and quick operational reviews.
Initial stability affects comfort, safety, and roll response. A larger positive GM usually gives a stronger restoring tendency at small angles. A very small GM suggests tender behavior. A negative GM indicates instability in the upright condition. Engineers use these checks while reviewing loading conditions, ballast changes, cargo shifts, or deck equipment additions. The relation is also helpful for teaching naval architecture basics because it links geometry to physical behavior in a simple way.
This shortcut is most reliable at modest heel. It is not a full replacement for complete stability analysis. Real vessels show changing waterplane geometry, free surface effects, and non-linear righting arms. Those effects grow as heel increases. That is why larger angles need cross curves, hydrostatic tables, or software-generated stability books. Even so, the quick G-M angle method remains valuable. It supports screening studies, sanity checks, and engineering communication. Used with clear assumptions, it can save time and highlight whether a case deserves deeper investigation.
GM is metacentric height. It is the vertical distance between the center of gravity and the metacenter. A positive value supports upright stability at small heel angles.
GZ is the righting arm. It is the horizontal lever between buoyancy and weight lines during heel. A larger lever creates a larger restoring moment for the same displacement.
The relation is strongest at small heel angles. It is commonly used for initial stability checks. For larger heel, use full stability curves or hydrostatic data for better accuracy.
You may enter different displayed units, but the calculator converts both internally before solving. The physical dimensions must still represent the same length quantity.
If GZ exceeds GM in the direct angle mode, the sine ratio becomes greater than one. That is not valid for this simplified relation. Input ranges must remain physically consistent.
Righting moment is the restoring turning effect. A simple estimate multiplies displacement or weight by GZ. This gives a quick measure of restoring strength in kN·m when weight is entered in kN.
Yes, for early engineering checks. It can support quick reviews of ships, barges, and floating units. Final approval should still rely on detailed stability documentation and governing standards.
The graph shows how righting arm changes with heel for the calculated GM. It helps compare sensitivity, inspect trend shape, and communicate stability behavior during design discussions.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.