Ship Stability (Metacentric Height) Calculator: Metacentric Height, Free Surface & Angle of Loll

Ship Stability (Metacentric Height) Calculator

GM = KM − KG · free surface correction · stiff vs tender

Rectangular Hull Approximation

A true box: the waterplane is treated as a plain rectangle and the underwater body as a plain prism. Real hull forms taper towards the bow and stern, so treat every figure below as an order-of-magnitude teaching estimate, not a hull-specific result.

Beam is cubed in the waterplane moment — it drives BM harder than anything else here.
1.00 = true box. Lower values only shrink the displaced volume; the rectangular waterplane assumption behind I stays in force either way.
Salt water ≈ 1.025, fresh water ≈ 1.000.
For a true box the two agree exactly. They diverge once Cb drops below 1.

Data From Your Stability Booklet ★

Real hydrostatic data always overrides the box approximation. Enter whatever your booklet or hydrostatic curves actually give you.

Used only for the rolling period and the stiff/tender judgement — clear it to skip those.

Loading Condition

The only quantity here you control by how you load. Everything else is fixed by the hull and the draft.
Auto-filled from volume × density in box mode. Override with the booklet figure — free surface correction divides by this.

Free Surface (Slack Tanks)

Any tank that is neither pressed full nor empty carries a free surface. The liquid slides to the low side as the vessel heels and G effectively rises, costing GM — and the loss depends on the tank's shape, not on how much liquid is in it.

Result

Enter the hull dimensions, the draft and where the centre of gravity sits, then add any slack tanks — you will get BM, KM, the solid and fluid GM, an angle of loll if the vessel is unstable, and a stiff-versus-tender reading of how it will behave in a seaway

Everything above rests on a rectangular hull with a rectangular waterplane, which no real vessel has. A genuine hull narrows towards the ends, so its waterplane second moment is smaller than the box figure and BM comes out lower — the box approximation is therefore optimistic about stability, which is the wrong direction to be wrong in. KB from the box or Morrish estimate is likewise only an approximation of a real vertical centre of buoyancy, and both assume upright, even-keel floating with no trim, no list and no hull deflection. GM describes initial stability at small angles only: it says nothing about the righting lever curve at larger heel, the range of positive stability, the downflooding angle or the effect of deck-edge immersion, all of which decide whether a vessel actually survives a knockdown. Free surface correction assumes each tank is a plain rectangle with an unrestricted surface; swash bulkheads reduce the effect substantially and irregular tank shapes change it. The commonly quoted minimum GM figure is informational context, not a regulatory determination — actual criteria depend on the vessel type, the flag state and the applicable intact stability code, and are assessed on the full righting lever curve rather than GM alone. Use this to build intuition and to check the shape of an answer, never to make a loading, ballasting or departure decision.

GM Calculator: Metacentric Height, Free Surface & Angle of Loll

GM, the metacentric height, is the single number that decides whether a ship rolls back upright after it heels or just keeps going over, and this metacentric height calculator builds it the same way a stability booklet does, starting from BM, then KM, then finally GM. If you don’t have real hull data, a box-hull mode gets you a rough teaching estimate from length, beam, draft and block coefficient, beam matters more then anything else here because the waterplane moment scales with beam cubed, widen a hull by just a tenth and BM jumps by roughly a third. Where a lot of basic calculators fall short is free surface, this one doesn’t, a half-empty tank shifts it’s own liquid as the ship heels, and that virtual rise in G eats into your GM regardless of how much liquid is actually sitting in the tank, so a slack tank can genuinely be more dangerous then a full one. Beyond the raw number, it interprets what your GM actually means, a negative value settles into an angle of loll rather then flipping straight over, and even a healthy positive GM can be too much of a good thing, since a stiff ship snaps back so hard it can injure crew and shift cargo, while a tender one rolls slow with little margin left in reserve. You can line up 2 to 4 loading conditions side by side too, light ship against fully loaded against something in between with slack tanks.

How to Use

Step 1: Pick your data source
  • Choose Box-hull estimate if you don’t have real hull data and just want a rough teaching figure, or My own hydrostatics if you’ve got actual numbers from a stability booklet or hydrostatic curves.
Step 2a: Using the box-hull estimate
  • Enter your Length on Waterline and Beam, remember beam is cubed in this calculation so it drives the result harder then anything else.
  • Enter your Draft and Block Coefficient, 1.00 means a true rectangular box, lower values only shrink the volume, they don’t change the flat-sided waterplane assumption underneath.
  • Enter your Water Density, roughly 1.025 for salt water, 1.000 for fresh.
  • Pick your KB Estimate Method, the simple Box method (T ÷ 2) or the Morrish/Normand approximation, these two agree exactly for a true box and only start to diverge once your block coefficient drops below 1.
Step 2b: Using your own hydrostatics instead
  • Choose What do you already have?: KM directly from your hydrostatic curves, KB and BM separately, or KB along with waterplane I and volume V. Whichever fields match your choice appear below, fill them in with your real numbers.
  • Optionally enter Beam, used only for the rolling-period and stiff/tender read later, clear it if you’d rather skip that part.
Step 3: Set your loading condition
  • Give this condition a Name, like “Fully loaded” or “Light ship”, so you can tell it apart later if your comparing several.
  • Enter your KG, the height of the centre of gravity above the keel, this is the one number here that actually depends on how you load the ship, everything else comes from the hull and the draft.
  • Check the Displacement field, it auto-fills from volume times density in box mode, but override it with your booklet figure if you have one, since the free surface correction divides by this number.
Step 4: Add any slack tanks
  • Choose your Tank Input Method: from rectangular tank dimensions or free surface moments directly if you already have them calculated.
  • Tap Add Tank for every tank that’s neither completely full nor completely empty, remember, the free surface loss depends on the tank’s shape, not on how much liquid is actually inside it.
Step 5: Calculate and read your result
  • Tap Calculate GM. You’ll see BM, KM, solid GM and fluid GM after the free surface correction, an angle of loll if the vessel comes out unstable and a stiff-versus-tender read based on the rolling period.
Step 6: Compare more then one condition
  • Tap Add to Comparison to save this condition, then change your inputs and calculate again for a different one, light ship, fully loaded or a partly-filled condition with slack tanks, they’ll line up side by side. Use Clear Comparison to start fresh.
Step 7: Export
  • Use Print / PDF for a clean printable copy or Copy to paste the numbers into your notes.

Key Features

  • Full GM chain, BM from I ÷ V, then KM, then GM, either from a box-hull estimate or your own real hydrostatic data
  • Free surface correction from actual tank geometry, something most basic GM calculators skip entirely
  • Angle of loll, calculated automatically whenever GM comes out negative
  • Stiff versus tender interpretation, using the GM/beam ratio and estimated rolling period, not just a bare number
  • 2 to 4 loading conditions compared side by side, light ship, loaded, partly full with slack tanks
  • Two KB estimate methods, simple box or Morrish/Normand, useful for seeing how much they diverge as block coefficient drops
  • A mandatory, prominent safety disclaimer kept right at the top, never buried or collapsed

Formula / Logic Used

The GM Chain

BM=IV,KM=KB+BM,GM=KM−KGBM = \frac{I}{V}, \qquad KM = KB + BM, \qquad GM = KM – KG

Box-Hull Waterplane Moment

I=L×B312I = \frac{L \times B^3}{12}

Beam is cubed here, which is exactly why widening a hull even slightly has such an outsized effect on stability compared to changing it’s length.

Free Surface Correction

FSM=i×ρliquid,FSC=∑FSMΔ,GMfluid=GMsolid−FSCFSM = i \times \rho_{liquid}, \qquad FSC = \frac{\sum FSM}{\Delta}, \qquad GM_{fluid} = GM_{solid} - FSC

Where iii is each tank’s own free-surface second moment of area, this loss happens regardless of how full the tank is, as long as it’s slack, since it’s the tank’s shape that matters, not the volume of liquid inside it.

Angle of Loll (When GM is Negative)

tan⁡2θ=−2×GMBM\tan^2\theta = \frac{-2 \times GM}{BM}

Rolling Period (Stiff vs Tender)

T=2πkg×GM,k≈0.35BT = \frac{2\pi k}{\sqrt{g \times GM}}, \qquad k \approx 0.35B

A shorter period means a stiffer, more violent roll, a longer period means a tender, slower roll with less stability margin in reserve.

Who Should Use This Tool

Naval architecture and marine engineering students learning the GM chain, free surface effect and stiff/tender vessel behaviour for a ship stability course. This tool is explicitly educational, real loading, ballasting or departure decisions must always go through the vessel’s own approved stability booklet or a qualified naval architect.

Frequently Asked Questions (FAQs)

1. How is a ship’s metacentric height (GM) calculated?

Divide the waterplane’s second moment of area by displaced volume to get BM, add the height of the centre of buoyancy to get KM, then subtract the height of the centre of gravity to get GM. This tool builds that exact chain, either from a rough box-hull estimate or from your own real hydrostatic data.

2. Why does a half-empty tank affect stability more than people expect?

As the ship heels, liquid in a slack tank slides toward the low side, and that shift effectively raises the centre of gravity, this loss depends entirely on the tank’s shape and surface area, not on how much liquid is actually in it. That’s why a half-full tank can genuinely hurt stability more then a completely full or completely empty one, and this tool calculates that correction from real tank dimensions rather then skipping it.

3. What is an angle of loll, and how is it different from capsizing?

When GM is negative, the vessel is unstable upright but can actually settle into a stable resting angle to one side, called the angle of loll, rather then continuing to roll over. It’s still a genuine emergency though, since the ship has no positive stability at zero heel, and correcting it wrongly, like filling a high tank, can make things considerably worse rather then better.

4. What does it mean if my calculated GM is very high?

A large GM makes a ship “stiff,” meaning it snaps back upright quickly and violently after a roll, which can be uncomfortable, and even dangerous, throwing cargo lashings and crew around, despite technically being very stable. This tool’s rolling period calculation helps show whether a vessel’s fine or genuinely too stiff for comfortable and safe operation.

5. Can I use this tool’s numbers for an actual voyage or loading decision?

No, absolutely not, this tool exists purely for building intuition and checking the rough shape of an answer using a simplified box-hull approximation that no real vessel actually has. Every real loading, ballasting, or departure decision must go through the ship’s own approved stability booklet, hydrostatic curves and a qualified naval architect’s sign-off.

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