Buoyancy Stability Calculator

Why Metacentric Height Matters

Stability determines whether a vessel can shrug off waves or tip perilously in choppy water. The metacentric height, usually abbreviated as , is the first metric naval architects check when assessing a new design. It describes the distance between the center of gravity and the metacenter—the point about which a tilted hull instantly rotates. A positive value means the ship produces a righting moment that pushes it upright after a small disturbance. Negative values indicate danger: the center of gravity sits higher than the metacenter, so a nudge can escalate into a capsize. Recreational skippers and professional mariners alike rely on quick estimates of to determine if a given loading plan is safe before leaving port.

This calculator provides a browser-based approximation for small craft. By entering beam width and the vertical locations of the center of gravity and center of buoyancy, you receive an estimate of initial stability. The tool intentionally keeps the math approachable so that boat builders, sailing coaches, and curious students can experiment with hull proportions without specialized software. Results are not a substitute for a professional inclining experiment, yet they demystify a concept that often appears in naval architecture textbooks without a practical walkthrough.

The Underlying Physics

Archimedes’ principle tells us that a floating body displaces a volume of water equal to its own weight. The center of that displaced volume is the center of buoyancy, . When the boat heels slightly, the shape of the underwater volume changes and the new center of buoyancy shifts sideways, forming a line with the original center of gravity, . The intersection of this line with the centerline creates the metacenter . The distance between and is the metacentric height.

The simple equation implemented here assumes a rectangular waterplane and uses draft approximated as half the beam. Under those assumptions the transverse metacentric height can be estimated with

The first term represents the vessel’s geometric moment of inertia divided by the displaced volume. Subtracting the distance from keel to the center of gravity () and adding the height of the center of buoyancy () adjusts this geometric value to the actual weight distribution. Although real hulls have curved waterlines and varying drafts, the equation captures how wider beams and lower centers of gravity increase stability.

Righting moment—the torque that pulls a boat upright—equals the vessel’s displacement times times the sine of the heel angle. In MathML:

Here denotes displacement and the heel angle in radians. Doubling doubles the righting moment for a given heel. That is why naval architects focus on keeping low and spreading hull form outward.

Step-by-Step Example

Consider a 3 m beam sport-fishing skiff. Gear and crew raise the center of gravity to 0.9 m above the keel. Hydrostatic analysis shows the center of buoyancy at 0.45 m. Plugging these into the calculator, assuming a 1.5 m draft, yields:

A metacentric height of 0.30 m falls within the comfortable range for small craft. If the skipper stores a heavy cooler on the cabin roof, raising to 1.3 m, the computation becomes:

This negative value signals that the boat would capsize easily. Simply moving the cooler to the deck lowers and restores stability. The example demonstrates how small shifts in weight can have outsized effects on righting moment.

Comparison of Hull Types

The table below contrasts typical ranges for different vessel categories, assuming moderate loading:

Vessel Type	Typical Beam (m)	Typical (m)
Dinghy	1.5	0.2 – 0.4
Sailboat Cruiser	3.5	0.5 – 1.2
Trawler	4.5	1.0 – 1.8
Catamaran	6.0	2.0 – 3.5

Higher values indicate a stiffer ride. Racing dinghies purposely keep low for responsiveness, while catamarans rely on very wide beams to achieve massive righting moments.

Limitations and Assumptions

Because the calculator assumes draft equals half the beam and a rectangular waterplane, results should be viewed as rough estimates. Real hulls feature complex curvature, appendages, and ballast arrangements that influence stability. Free-surface effects from partially filled tanks can raise the effective center of gravity. Likewise, the formula addresses only initial stability at small heel angles; it cannot predict behavior in a knockdown or broach where nonlinear hydrodynamics dominate. Before venturing offshore, captains should consult classification society rules and conduct inclining tests under professional supervision.

The tool also expects non-negative numeric inputs. Extreme values, such as a center of gravity below the keel, may produce unrealistic results. Use common sense when interpreting the output and double-check measurements when converting from imperial units to meters.

Practical Tips for Enhancing Stability

When a quick calculation reveals marginal stability, several design tweaks can improve . Adding ballast low in the hull or keel lowers . Widening the beam or installing outriggers increases the moment of inertia. For cargo vessels, careful loading plans that distribute weight evenly fore and aft prevent trim issues that compromise safety. Recreational sailors often stow heavy gear like batteries under the cockpit sole to keep the center of gravity low without reducing cabin space.

Electronics can assist as well. Modern stability monitors measure heel angle and acceleration, warning crews when thresholds approach dangerous levels. Recording these data alongside calculator estimates builds confidence in the vessel’s capabilities and highlights when maintenance—such as replacing waterlogged foam or saturated wood—is required.

Worked Example in Detail

To illustrate a full workflow, imagine converting a fishing skiff into a dive platform. The beam measures 2.8 m. After adding tanks and gear, the center of gravity rises to 1.2 m, while hydrostatic tests place the center of buoyancy at 0.6 m. Draft is approximated as 1.4 m. Plug the numbers into the calculator:

The first fraction evaluates to roughly 0.47. Subtracting 1.2 and adding 0.6 leaves ≈ -0.13 m. The negative outcome signals that the modifications made the boat dangerously top-heavy. Options include removing weight from the roof, installing a heavier keel, or widening the hull with detachable pontoons. Once becomes positive—perhaps 0.4 m after adjustments—the owner can plan dive excursions with greater confidence.

Related Tools

For broader design studies, pair this calculator with the Sailboat Hull Speed Calculator to estimate performance and the Deep Sea Pressure Hull Thickness Calculator when evaluating structural requirements for submersible projects. Combining these tools paints a fuller picture of how geometry, materials, and weight distribution affect marine engineering.

Conclusion

The metacentric height encapsulates the delicate balance between buoyancy and gravity that keeps vessels upright. By experimenting with beam width and weight placement in this calculator, you gain intuition about how design choices influence safety. Use the estimates as a springboard for more detailed analysis and never neglect official stability guidelines, especially when carrying passengers or venturing offshore.