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Boat metacentric height stability

With a GM of 0.5 m, a displacement of 1000 tonnes and 5° of heel, the righting lever GZ is 0.043577871374 m and the righting moment 43.577871373829 tonne-metres. Check initial stability without a full hydrostatics package. Free in your browser.

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Worked example

If you enter

Gm: 0.5
Displacement: 1,000
Heel angle: 5

You get

Metacentric height: 0.5
Heel rad: 0.0872664626
Sin heel: 0.0871557427
Righting lever: 0.0435778714

Initial transverse stability of a floating vessel is often summarised by the metacentric height GM and the righting moment that appears when the hull heels a few degrees from upright. Naval students, yacht designers, surveyors, and voyage planners need a quick deterministic check that converts known GM, displacement, and a chosen heel angle into the righting lever GZ and the restoring moment without opening a full hydrostatic package. This boat metacentric height stability calculator accepts those three scalars, applies the classical small-angle identities GZ equals GM times sine of heel and righting moment equals displacement times GZ, and returns rounded levers, moments, sine of heel, heel in radians, and a stability class from the sign of GM. Positive GM means the vessel is initially stable; zero is neutral; negative is initially unstable.

What metacentric height and righting moment mean

When a ship or boat floats upright, the centre of buoyancy B sits vertically under the centre of gravity G. A small heel angle φ shifts the underwater volume so B moves transversely to a new position; the vertical through the heeled buoyancy intersects the upright centreline at the metacentre M. The vertical distance from G to M is the metacentric height GM.

For small heels the righting lever—the horizontal offset between the lines of weight and buoyancy—is GZ equals GM times the sine of the heel angle. Multiplying that lever by the vessel displacement Δ yields the righting moment that tends to restore the hull when GM is positive, or to increase the heel when GM is negative. Displacement here is the weight of the craft (or a force-equivalent mass unit used in naval architecture tables, such as tonnes), not the linear displacement of kinematics.

The model is the textbook initial-stability idealisation: it assumes wall-sided or otherwise well-behaved sections over a modest heel, ignores free-surface moments and wind heeling unless you fold them into GM or Δ yourself, and is most trustworthy from a few degrees up to about ten degrees rather than near capsize. Prefer a full GZ curve or large-angle hydrostatic tool when the problem is dynamic stability, flooding, or angles far beyond the linear range.

What you can do with it

Naval architecture homework

Students enter GM, displacement, and a small heel from textbook problems and verify GZ and righting moment against hand calculations.

Small-craft design checks

Designers plug a known loaded GM and displacement to estimate restoring moment at five or ten degrees for early concept reviews.

FAQ

What heel angle should I use for initial stability?

Initial-stability theory is most valid for small heels, often a few degrees up to about 10°. Absolute heel must not exceed 90° for this model.

What if GM is negative?

Negative GM is accepted and classified as unstable. The righting lever and moment reverse sign, indicating a moment that increases heel rather than restoring upright.

Why must displacement be positive?

Displacement is the vessel weight (or force-equivalent mass) used as Δ in RM = Δ·GZ.

Which units should I use?

Any consistent set. GM and GZ share one length unit, such as metres, and the righting moment comes out in displacement times length, for example tonne-metres or newton-metres. Mixing units gives a moment that looks valid but is not.

When do I need a full stability curve instead?

Beyond small heel angles. This initial-stability model is most trustworthy from a few degrees up to about 10°; for large angles, flooding, dynamic stability, liquids sloshing in tanks or wind, use a full GZ curve or a hydrostatic tool.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/travel/metacentric-stability

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/travel/metacentric-stability \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"gm":0.5,"displacement":1000,"heel_angle":5}'

const res = await fetch("https://api.kit.forhosting.com/travel/metacentric-stability", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "gm": 0.5,
    "displacement": 1000,
    "heel_angle": 5
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/travel/metacentric-stability",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "gm": 0.5,
    "displacement": 1000,
    "heel_angle": 5
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/travel/metacentric-stability", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"gm":0.5,"displacement":1000,"heel_angle":5}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"gm":0.5,"displacement":1000,"heel_angle":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/travel/metacentric-stability", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "gm": 0.5,
  "displacement": 1000,
  "heel_angle": 5
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "travel.metacentric_stability",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

Invalid displacement or missing fields fail with invalid_input and are not billed.

Input contract, response shape, and validation rules

Non-positive displacement is rejected because a zero or negative ship weight is outside the domain of the restoring-moment formula. Heel beyond plus or minus ninety degrees is rejected for this initial-stability model so sine arguments stay in a sensible upright-to-beam range. Example: GM equals zero point five, displacement equals one thousand, heel equals five degrees produces GZ equal to zero point five times sin five degrees and RM equal to one thousand times that lever. POST a JSON body with three required finite numbers: gm (aliases metacentric_height, GM, metacentric), the metacentric height in a length unit such as metres; displacement (aliases delta, disp, displacement_force, Delta), a strictly positive force or force-equivalent mass consistent with the moment unit you want; and heel_angle (aliases heel, phi, angle_deg, heel_deg), the heel from upright in degrees with absolute value not greater than ninety. Each field may arrive as a JSON number or as a strict numeric string; missing fields, empty strings, NaN, Infinity, and non-numeric text fail with invalid_input naming the field. On success the body echoes rounded gm, metacentric_height, displacement, and heel_angle; adds heel_rad and sin_heel; reports righting_lever and gz (identical values); reports righting_moment; sets stability_class to stable, neutral, or unstable from the sign of GM; and includes formula, method, unit_note, and a short message string for display. Pricing is a flat $0.002 per successful item call; validation failures do not consume balance.

Where a boat metacentric stability API earns its keep

Maritime academies and yacht-design courses use a boat metacentric height stability calculator so learners can check hand-worked GZ and RM at assigned heels without spreading unit errors across a spreadsheet. Surveyors and small-craft designers keep a known lightship or loaded GM from an inclining experiment and want a fast moment estimate at five or ten degrees for briefing notes. Keep units consistent—metres with tonnes for tonne-metres, or metres with newtons for newton-metres—and treat negative GM as an educational unstable case rather than a silent absolute value. For large-angle stability, free-surface correction, or wind heeling moment balance, compose this tool with your own data or a specialised hydrostatic package; this endpoint only evaluates the classical initial-stability identities at the heel you supply.

Integration use cases

Training simulators and unit tests

Generate deterministic stability fixtures—including unstable negative-GM cases and invalid_input when displacement is non-positive—for CI and LMS auto-graders.

Technical questions

How do I calculate boat righting moment from metacentric height?

Send finite gm (GM), positive displacement Δ, and heel_angle in degrees. The API returns GZ = GM·sin(φ) and righting_moment = Δ·GZ with consistent units.

What does the boat metacentric height stability calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

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