Buoyancy & Stability (GM) Calculator

Important: This tool provides an educational, first-pass estimate of initial transverse metacentric height (GM) for small heel angles. It does not replace a stability booklet, class rules, an inclining experiment, or professional naval architecture review. Vessel stability is safety-critical; use this calculator to build intuition and to sanity-check inputs, not to certify seaworthiness.

What this calculator does

When a vessel heels slightly, buoyancy shifts sideways and creates a righting (or overturning) moment. For small angles, the “initial stiffness” of that response is commonly summarized by the metacentric height:

• GM > 0 generally indicates positive initial stability (a restoring tendency for small heel angles).
• GM ≈ 0 suggests neutral initial stability (tender, slow to return, sensitive to loading changes).
• GM < 0 indicates initial instability (small heels tend to increase rather than self-correct).

This calculator estimates GM from three inputs:

• Beam, B (m): the vessel’s breadth at the waterline region relevant to initial stability (often approximated by maximum beam for simple checks).
• Center of Gravity above Keel, KG (m): vertical position of the vessel’s total weight (hull + machinery + payload + fluids) measured up from the keel/baseline.
• Center of Buoyancy above Keel, KB (m): vertical position of the centroid of displaced volume (the buoyant force line of action) measured up from the same reference.

Datum consistency matters: KG and KB must be measured from the same vertical reference (typically the keel/baseline). If you mix reference points (e.g., KG from the waterline and KB from the keel), the computed GM will be meaningless.

Core formulas (initial transverse GM)

Initial transverse metacentric height is often written as:

GM = KM − KG, where KM = KB + BM.

The term BM (metacentric radius) depends on waterplane geometry and displacement:

• BM = I_T / ∇
• I_T = transverse second moment of area of the waterplane about the centerline
• ∇ = displaced volume (volume of water displaced)

This calculator uses a simplified “boxy waterplane” approximation that is commonly used for quick, rough checks when detailed hydrostatics are not available. In its simplest form, if one assumes a rectangular waterplane and a coarse relationship between beam and draft, BM can be approximated as proportional to beam. A frequently used back-of-envelope approximation is:

BM ≈ B² / (12·d) and with a crude draft estimate d ≈ B/2, this reduces to BM ≈ B/6.

So the calculator’s simplified estimate becomes:

GM ≈ (KB + B/6) − KG

MathML (for reference)

Units: If B, KG, and KB are in meters, GM is in meters.

How to interpret the result

GM is an initial stability indicator. It describes the slope of the righting-arm curve near zero heel and is most meaningful at small angles (often the first few degrees). A larger positive GM typically means the vessel is “stiffer” (returns upright more strongly for small angles), while a small positive GM indicates a “tender” vessel that may feel slow and roll more deeply.

However, “good” GM is not a universal number. The same GM might be acceptable for one vessel type and uncomfortable or unsafe for another depending on:

• Vessel type (workboat, sailboat, barge, ferry, yacht)
• Loading condition (cargo height, passengers, tanks, deck loads)
• Sea state and operational profile
• Free-surface effects in slack tanks (can materially reduce effective GM)
• Downflooding points and reserve buoyancy (large-angle behavior)

Worked example (step-by-step)

Suppose you have a small craft and you estimate/measure the following (all from the keel/baseline):

• Beam B = 3.0 m
• Center of buoyancy KB = 0.60 m
• Center of gravity KG = 1.20 m

1. Compute approximate BM: BM ≈ B/6 = 3.0/6 = 0.50 m
2. Compute KM: KM = KB + BM = 0.60 + 0.50 = 1.10 m
3. Compute GM: GM = KM − KG = 1.10 − 1.20 = −0.10 m

Interpretation: A negative GM in this simplified check suggests the vessel may be initially unstable in that loading condition (or that the simplified geometry is not representative of the actual waterplane/displacement). Practically, you would treat this as a red flag: verify KG and KB, confirm the reference datum, consider tank free-surface, and obtain proper hydrostatic/stability analysis before operation.

Quick comparison table (context, not a rulebook)

The table below is a qualitative guide to help interpret what the computed GM might imply. It is not a safety standard and should not be used to declare a vessel “safe” or “unsafe.”

Assumptions and limitations (read before using)

• Small-angle only: GM characterizes initial stability near upright. It does not describe large-angle righting energy, range of positive stability, or capsize resistance.
• Simplified hull geometry: The BM estimate uses a coarse approximation (boxy/rectangular waterplane and a rough draft relationship). Real waterplanes (flare, tumblehome, chines, round bilges) can change I_T, ∇, and thus BM materially.
• No displacement or waterplane input: Because displacement volume ∇ and waterplane inertia I_T are not entered, the tool cannot compute BM from first principles for your exact condition.
• Free-surface effects not included: Partly filled tanks can reduce effective GM (sometimes dramatically). This calculator does not account for free-surface moments.
• No downflooding/edge immersion check: Deck edge immersion, openings, and downflooding points can dominate real-world safety but are outside the scope here.
• Static estimate: Waves, wind heel, turning, passenger motion, and dynamic stability effects are not modeled.
• Datum consistency required: KG and KB must be measured from the same baseline (typically keel). If unsure, stop and reconcile references before trusting any output.

Tips to improve your inputs

• If you have hydrostatics, use the published KM (or BM and KB) for the relevant displacement and trim rather than relying on the B/6 approximation.
• For KG, ensure you are using the current loading condition (fuel, water, cargo, passengers) and a consistent vertical datum.
• If tanks are slack, consider doing a separate free-surface correction with proper tank geometry.

References (for deeper study)

• Eugene A. Comstock (ed.), Principles of Naval Architecture, SNAME.
• Edward V. Lewis (ed.), Principles of Naval Architecture: Stability and Strength, SNAME.