About Metacentric Height Calculator
The metacentric height calculator evaluates the transverse intact stability of a floating box-shaped body such as a barge, pontoon or a rectangular idealisation of a ship hull. It computes the displaced volume and displacement, the height of the centre of buoyancy above the keel KB, the metacentric radius BM, the height of the transverse metacentre KM, and the metacentric height GM = KB + BM - KG, which governs whether the vessel is stable.
Enter the waterline length L, beam B, draft T, the vertical centre of gravity above the keel KG, and a heel angle, all in consistent SI units (metres). For a wall-sided box the calculator uses KB = T/2 and BM = I/V = B^2/(12T), reports the metacentric height and stability verdict, and evaluates the small-angle righting arm GZ = GM sin(theta) and righting moment RM = W x GZ.
How It Works
- Enter the box dimensions L, B and T (m) and the centre of gravity height KG (m) above the keel.
- The displaced volume is V = L x B x T and the displacement is rho x V, with seawater density 1025 kg/m^3 by default.
- KB is taken as T/2, the transverse waterplane second moment of area is I = L x B^3 / 12, and BM = I / V = B^2 / (12 T).
- The metacentre is KM = KB + BM and the metacentric height is GM = KM - KG. The body is stable when GM > 0.
- For the entered heel angle the righting arm is GZ = GM sin(theta) and the righting moment is RM = W x GZ.
Worked Example
A rectangular box barge has length L = 20 m, beam B = 8 m and draft T = 2 m, with the centre of gravity at KG = 3 m above the keel, floating in seawater (rho = 1025 kg/m^3). The displaced volume is V = 20 x 8 x 2 = 320 m^3, giving a displacement of 1025 x 320 = 328,000 kg = 328 t and a weight W = 328000 x 9.81 / 1000 = 3217.68 kN. The centre of buoyancy is KB = T/2 = 1.0 m. The metacentric radius is BM = B^2 / (12 T) = 64 / 24 = 2.667 m, so KM = KB + BM = 3.667 m and GM = KM - KG = 3.667 - 3 = 0.667 m. Because GM > 0 the barge is stable. At a 5 degree heel the righting arm is GZ = 0.667 x sin(5 deg) = 0.0581 m and the righting moment is RM = 3217.68 x 0.0581 = 187 kN.m.
Formulas
- Displaced volume (box)
V = L * B * T- Centre of buoyancy above keel (box)
KB = T / 2- Metacentric radius
BM = I_waterplane / V = B^2 / (12 * T)- Metacentric height
GM = KM - KG = KB + BM - KG- Righting arm and moment
GZ = GM * sin(theta) ; RM = W * GZ
Standards & References
- Principles of Naval Architecture (SNAME)
- Ship hydrostatics and stability (Biran & Lopez-Pulido)
- Basic Ship Theory (Rawson & Tupper)
- IMO Intact Stability Code (IS Code) concepts
Frequently Asked Questions
What does the metacentric height GM tell me?
GM is the distance between the centre of gravity G and the transverse metacentre M. A positive GM means the body is statically stable and will return upright after a small heel; a negative GM means it is unstable and will capsize or loll. GM also sets the initial stiffness and roll period of the vessel.
Why is BM equal to B^2 / (12 T) for a box barge?
The metacentric radius is BM = I / V, where I is the transverse second moment of area of the waterplane and V is the displaced volume. For a rectangular waterplane I = L B^3 / 12 and V = L B T, so BM = (L B^3 / 12) / (L B T) = B^2 / (12 T). Beam dominates stability because it enters squared.
Does the calculator handle real ship hull shapes?
No. It uses the box-barge idealisation with a wall-sided, rectangular waterplane and KB = T/2, which is accurate for pontoons and barges and a useful first estimate for boxy hulls. For a ship-shaped hull KB and the waterplane inertia must come from the actual hydrostatic curves or offsets.
Is the righting arm formula valid at large heel angles?
GZ = GM sin(theta) is the small-angle (metacentric) approximation and is reliable up to roughly 7 to 10 degrees of heel. Beyond that the metacentre moves and a full large-angle stability (cross-curves) analysis is required, so treat the plotted GZ curve here as the initial-stability trend only.