GZ and KN Curve Calculations: Formula and Worked Examples

What do GZ and KN mean?

• GZ is the righting lever: the perpendicular distance between the lines of action of the ship’s weight and buoyancy. It is a distance in metres, not a moment.
• KN is the corresponding lever measured from K: the perpendicular distance from the keel datum K to the line of action of buoyancy at the selected heel.
• KG is the height of G above K. Use the same K datum as the KN data.
• θ (theta) is the angle of heel. Use degree mode when the question gives angles in degrees.

KN is a hydrostatic value for the vessel’s form and stated calculation assumptions. Changing KG at the same displacement and with the same KN data changes GZ.

The GZ and KN formula

Technical reference: MCA example stability booklet, KN tables and GZ relationship.

How to interpolate a KN value

If the required displacement lies between two tabulated values, interpolate at the same angle of heel. Suppose the table gives these KN values at 20°:

1. Interpolation factor = (10,000 − 9,000) ÷ (11,000 − 9,000) = 0.50.
2. Interpolated KN = 2.60 + 0.50 × (2.80 − 2.60) = 2.70 m.
3. GZ at 20° = 2.70 − 7.00 × sin 20° = 2.70 − 2.39414 = 0.306 m.

If KN is presented as cross-curves, use the required displacement and read the curve for each heel angle, checking the axis labels. Where a question also requires interpolation between heel angles, follow its stated method and calculate sin θ for the actual angle required.

Check the heading before using cross-curves. Some older data give righting levers for an assumed KG rather than KN. Those values require a correction for the difference between assumed and actual KG; they must not be treated as KN.

Build the GZ table and plot the curve

Keep displacement at 10,000 t and KG at 7.00 m. Using the following illustrative KN values, repeat the calculation at each angle:

Values are rounded for display; calculations use unrounded sine values. This is invented teaching data, not Hindship or an approved vessel dataset.

• Horizontal axis: angle of heel in degrees.
• Vertical axis: righting lever GZ in metres.
• Positive GZ: a restoring tendency towards upright in this centreline-G example.
• Largest tabulated GZ: about 0.800 m at 40°. The exact curve maximum needs finer data around its peak.
• Vanishing positive stability: the curve crosses back through zero between 60° and 70° in this example. Negative GZ beyond that crossing acts to increase the heel.

Downflooding can occur before the curve reaches zero. A positive GZ at one angle does not establish that the loading condition meets all stability requirements. AMSA’s intact stability standard illustrates why curve shape, range, areas and downflooding are assessed together.

How does free surface affect GZ?

If the question specifies a constant virtual-rise correction, add FSC to the solid KG before calculating GZ:

For the 30° example, suppose FSC = 0.10 m:

• KG_fluid = 7.00 + 0.10 = 7.10 m.
• Corrected GZ = 4.10 − 7.10 × 0.500 = 0.55 m.
• The loss of GZ at 30° is FSC × sin 30° = 0.05 m.

Do not simply subtract the full 0.10 m from every GZ ordinate. Under this method, the loss is FSC × sin θ.

A constant virtual rise is a specified calculation method, not a universal exact allowance at every heel angle. Actual liquid-shift corrections can vary with heel and tank geometry. Use the question’s instructions or the vessel’s approved method, including angle-specific GZ corrections where supplied. Do not apply a second correction to an already corrected KG or GZ.

See the MCA example’s solid/fluid KG calculation, and our Free Surface Effect page for the FSM and FSC calculation.

Area under a GZ curve: use radians

When the answer is required in metre-radians, convert the angle interval to radians. For Simpson’s first rule, use equally spaced ordinates and an even number of intervals.

From 0° to 40° in our table, there are four intervals of 10°:

Using the displayed rounded GZ values:

Area ≈ (0.174533 ÷ 3) × [0 + 4(0.144) + 2(0.306) + 4(0.600) + 0.800]

This is a Simpson estimate from the ordinates, not the exact area beneath the straight segments drawn above. It uses the uncorrected GZ table; recalculate the ordinates first if free surface must be included.

Common exam mistakes

• Writing GZ = KN − KG. The correction is KG × sin θ.
• Using the wrong displacement or trim table. Read the KN data headings and notes first.
• Using radian mode for an angle given in degrees. Check the calculator before evaluating sine.
• Calling KN the ship’s actual righting lever. You still need the KG correction.
• Using GZ ≈ GM × sin θ for the whole curve. This is an initial, small-angle approximation; at larger heel use suitable KN or other approved stability data.
• Subtracting FSC twice, or deducting FSC directly from every GZ. Follow the stated correction method.
• Forgetting radians in an area calculation. An area calculated using degree intervals is in metre-degrees; multiply by π/180 to convert it to metre-radians.

Short answers for revision

What is the difference between KN cross-curves and a GZ curve?

KN cross-curves supply hydrostatic levers across displacements and heel angles. A GZ curve uses the selected KN data and the loading condition’s KG, with the required corrections, to show righting lever against heel.

What happens if KG increases?

For unchanged KN data, a rise δKG reduces GZ by δKG × sin θ. For example, raising KG by 0.20 m reduces GZ at 30° by 0.10 m.

Can I use the basic formula when G is off the centreline?

An off-centre G introduces an additional transverse correction. The basic formula on this page assumes G is on the centreline; use the transverse-CG correction and sign convention specified in the problem.

Does a large GM guarantee a good GZ curve?

No. GM describes initial stability. Large-angle righting levers, curve area, range and downflooding must also be considered.