A real-time calculator for the single number that decides whether a ship rights itself or capsizes — the metacentric height GM. Adjust length, beam, draft and KG and see KB, KM, GM and the natural roll period update instantly, and build an intuition for ship stability.
Parameters
Ship length L
m
Length at the waterline
Beam B
m
Maximum waterline width. Drives BM (B cubed).
Draft d
m
Depth below the waterline. Drives KB ≈ d/2.
Centre-of-gravity height KG
m
Height of G above the keel. Set by cargo and ballast layout.
Midship section of a heeled ship. The centre of gravity G, centre of buoyancy B (shifted toward the immersed side), metacentre M, gravity and buoyancy force vectors, and the righting (or capsizing) moment are shown. The ship gently oscillates in roll.
GM vs centre-of-gravity height KG
Natural roll period vs GM
Theory & Key Formulas
$$BM=\frac{I_{wp}}{V},\qquad GM=KB+BM-KG$$
BM is the metacentric radius (B to M distance), I_wp the waterplane moment of inertia, V the displaced volume, KB the height of B above the keel, KG the height of G. The ship is stable when GM > 0.
$$T_{\text{roll}}=2\pi\,\frac{k}{\sqrt{g\,GM}}$$
The natural roll period scales as 1/√GM (k ≈ 0.35·B is an approximation for the radius of gyration). A large GM gives a short, stiff roll; a small GM a long, tender one.
What is the metacentric height?
🙋
Why does a ship right itself when it heels over a little? It's just floating on water — why is there any restoring force at all?
🎓
Great question. The trick is that the centre of buoyancy B shifts as the ship heels. When the ship stands upright, the centre of gravity G and the centre of buoyancy B both lie on the same vertical line. Heel the ship a bit to starboard and the underwater shape becomes asymmetric — more hull is submerged on the starboard side, so the centroid of the displaced volume (which is exactly B) moves toward starboard. G stays put because the structure has not moved. Now you have weight pulling down on the centreline and buoyancy pushing up a little to the right — together they form a couple that rotates the ship back upright. That couple is the righting moment.
🙋
OK, but then what is the metacentre M? The right panel shows KB, BM, KM, KG, GM all stacked up and I'm losing track of what is what.
🎓
Think of every symbol as "height above the keel K". KB is the height of B, KG is the height of G. BM is the distance from B up to M, given by BM = I_wp / V — a pure geometric quantity. M itself is defined as the point where the line of action of the heeled buoyancy force, extended upward, crosses the original ship centreline. As long as the heel angle stays small, M barely moves — it is a "near-fixed" point of the geometry. So KM = KB + BM is set by the hull, and GM = KM − KG tells you "by how much M sits above G". GM > 0 → M above G → righting couple → stable. GM < 0 → M below G → capsizing couple → over she goes.
🙋
So bigger GM is always better, right? When I drag KG down, GM climbs and the ship looks rock-solid in the animation.
🎓
Bigger is safer, yes — but only up to a point. Look at the period chart: as GM grows, the natural roll period T = 2π·k/√(g·GM) gets shorter. A stiff ship with huge GM whips back from heel almost instantly. The passengers feel a hard, jerky roll — they get seasick fast, and cargo on deck wants to break loose. The opposite extreme is a tender ship with very small GM: it sways slowly and feels comfortable, but in heavy seas the righting moment runs out and reserve stability is tiny. Naval architects aim for the sweet spot — about 0.3–0.5 m for cruise ships, 0.6–1.0 m for cargo ships, 1.0–1.5 m for tugs and fast warships.
🙋
Speaking of which — container ships stack containers really high on deck. That must push KG way up and crush GM. Is that safe?
🎓
You've put your finger on a key topic in naval architecture. Every container loaded on deck raises KG a little. So before sailing, the loading computer runs through every container's mass and position to recompute KG, and checks that GM still satisfies the IMO floor (0.15 m for passenger ships, separate rules for cargo). Pumping seawater into low ballast tanks pulls KG down and rescues GM. Fishing-vessel ice-accretion capsizes are the same physics in reverse — ice piles up on the superstructure, KG climbs, and GM goes negative. At speed, dynamic effects like parametric roll in head seas pile on top of all this, which is why the IMO's second-generation criteria go beyond a single static GM number to evaluate dynamic failure modes too.
Frequently Asked Questions
GM is the vertical distance from the ship's centre of gravity G up to the metacentre M, and it is the single most important number in basic ship stability. When the ship heels, the centre of buoyancy B shifts toward the immersed side; extending the line of action of the new buoyancy force upward gives the metacentre M. If M lies above G (GM > 0), gravity and buoyancy form a righting couple and the ship returns upright. If M lies below G (GM < 0) the couple becomes a capsizing couple. Passenger ships typically aim for GM = 0.3–1.5 m and cargo ships 0.6–1.0 m.
BM is the distance from the centre of buoyancy B up to the metacentre M, given by BM = I_wp / V. I_wp is the moment of inertia of the waterplane area about the longitudinal axis (L·B³/12 for a rectangular plan), and V is the displaced volume. Because BM scales with the cube of the beam B, wide vessels (catamarans, barges) have very large BM and high initial stability. BM is a purely geometric quantity, independent of the centre of gravity G.
A large GM creates a stiff ship with strong righting moments and a very short roll period T = 2π·k/√(g·GM). The vessel snaps back sharply (jerky rolls), which makes passengers seasick and can shift cargo. For cargo ships this risks load shifting; for passenger ships, severe discomfort. Conversely a small GM gives a tender ship with slow, lazy rolls but reduced reserve righting moment in heavy seas. Passenger ships typically target GM = 0.3–0.5 m to balance comfort and safety.
Adding weight high on deck raises the total centre of gravity KG, and since GM = KB + BM − KG, GM decreases. Filling low ballast tanks lowers KG and restores GM. In practice, a loading computer recomputes KG for every cargo configuration and checks GM against a regulatory floor (e.g. 0.15 m for passenger ships under IMO). Ice accretion (a classic capsize cause in fishing vessels) and high-stack container loading are evaluated with the same calculation.
Real-World Applications
Cargo loading and the loading computer: Container ships, bulkers and tankers recompute KG before every departure from the location of every container, every fuel tank and every ballast tank, then check that GM lands inside the regulatory and operational envelope. Too much GM means short roll periods and shifted cargo; too little means insufficient reserve stability for heavy weather. The bridge loading computer runs exactly this calculation in seconds and feeds back a safe stow plan to the crew.
Passenger-ship comfort design: Cruise ships deliberately use a modest GM (0.3–0.5 m) to stretch the natural roll period out to 10–15 seconds, producing a slow, soft motion. This also helps the ship avoid resonance with typical sea-state periods. Roll-damping tanks and fin stabilisers work alongside this tuning to make a stable platform for restaurants and pools. Slide GM in this tool from 2 m down to 0.4 m and you can see the period change dramatically.
Fishing and small craft capsize prevention: Ice accretion on northern fishing vessels, uneven hauls on purse seiners, and crowd-shifting on small passenger boats all reduce GM below the design assumption. Sufficient margin at the design stage plus a captain who can re-evaluate GM operationally is the heart of capsize prevention — and several IMO conventions exist specifically because GM dropped below zero in the wild.
Offshore structures and floating wind turbines: Semi-submersible offshore platforms and floating wind foundations need independent pitch- and roll-GM. Wide column spacing (large I_wp → big BM), mooring layout and deck-weight distribution decide stability. The same "BM = I_wp / V" geometric relation that this tool uses for ships applies directly to floater design.
Common Misconceptions and Pitfalls
The first big mistake is to treat the metacentre M as a fixed point of the ship. M is "essentially fixed" only at small heel angles. Past about 10°–15° the underwater section shape changes and M moves. This tool uses initial-metacentre, small-angle theory — the standard textbook framework for static stability — but for large heels (e.g. 30° roll in heavy seas) you need a full GZ righting-arm curve, GZ = GM·sinφ + ... , to evaluate finite-angle stability and the angle of vanishing stability. GM is an initial-stability number, not a guarantee about extreme rolls.
Next, "if static GM > 0 the ship cannot capsize" is not true. Real seas add waves, wind, free surfaces (liquid sloshing in tanks) and parametric roll, where head waves periodically change the effective GM. After several container-ship parametric-roll incidents in the 2000s, the IMO introduced the Second-Generation Intact Stability Criteria (SDC), covering five dynamic failure modes — pure loss of stability, parametric roll, dead-ship condition, broaching/surf-riding and excessive accelerations. GM is a necessary entry condition; an operational decision needs dynamic analysis.
Finally, do not assume the rectangular formula I_wp = L·B³/12 is enough. This tool uses it for clarity, but real ships have bulbous bows, fine sterns, camber and other curvature, so I_wp is found by integrating the actual waterline shape (numerically from the lines plan). A real I_wp is typically 0.6–0.85 of the box value. This tool is great for order-of-magnitude design decisions, but production naval-architecture work uses Bonjean curves and dedicated hydrostatics software (NAPA, GHS, ShipFlow) for accurate numbers.