Compute CWP directly from waterplane area, from station breadths using Simpson’s rule, or from polygon outlines.
AWP = — m²
CWP = —
AWP (Simpson) = — m²
CWP = —
AWP (polygon) = — m²
CWP = —
The waterplane coefficient describes how much of a rectangle defined by the ship's waterline length and maximum waterline breadth is occupied by the actual waterplane. The waterplane is the horizontal shape enclosed by the hull at a stated draft.
A waterplane with long, nearly parallel sides and full ends occupies a larger part of its reference rectangle and therefore has a higher CWP. A waterplane that narrows strongly towards the bow and stern has a lower value.
CWP is specific to the selected waterline. When draft, trim or heel changes, the waterplane geometry may also change. The coefficient should therefore be reported together with the condition at which it was calculated.
The coefficient is calculated from:
CWP = AWP ÷ (LWL × BWL)
| Symbol | Meaning | Usual unit |
|---|---|---|
| CWP | Waterplane area coefficient | Dimensionless |
| AWP | Area enclosed by the selected waterline | m2 |
| LWL | Length of the selected waterline | m |
| BWL | Maximum breadth on the same waterline | m |
Both the numerator and denominator are areas, so CWP has no unit. LWL, BWL and AWP must refer to the same waterline and loading condition.
Consider a vessel with:
LWL × BWL = 180 × 30 = 5,400 m2
CWP = 4,320 ÷ 5,400 = 0.800
The actual waterplane occupies 80.0% of the waterline reference rectangle.
Tons per centimetre immersion, normally written TPC, is the approximate mass required to increase a vessel's mean draft by one centimetre at the stated waterline.
For a small parallel change in draft:
TPC = ρ × AWP ÷ 100
AWP = 100 × TPC ÷ ρ
Here, ρ is water density in t/m3, AWP is in m2, and TPC is in t/cm.
The worked example gives the following seawater TPC when ρ = 1.025 t/m3:
TPC = 1.025 × 4,320 ÷ 100 = 44.28 t/cm
Entering 44.28 t/cm and a density of 1.025 t/m3 in the direct tab returns AWP = 4,320 m2 and CWP = 0.800.
Small-immersion assumption: TPC represents the local waterplane at the stated draft. For a large change in draft, the waterplane area may change materially and the result should be obtained from hydrostatic data or by integrating through the required draft range.
The calculator provides three methods. Each method first determines AWP and then divides it by LWL × BWL.
| Method | Main inputs | Area calculation |
|---|---|---|
| Direct | AWP, or TPC and water density | Entered directly or converted from TPC |
| Stations | Equally spaced full breadths or half-breadths | Simpson one-third rule or trapezoidal fallback |
| Polygon | Ordered waterplane boundary coordinates | Shoelace formula |
Use the direct mode when AWP is available from hydrostatic particulars, a hull model or another reliable source.
When only TPC is known, select the TPC input mode and enter the water density belonging to the same condition. The calculator converts TPC to waterplane area before calculating CWP.
The station method calculates waterplane area from breadths measured at equally spaced longitudinal positions over LWL.
The calculator accepts either:
With N station values, the number of intervals is N − 1 and the spacing is:
Δx = LWL ÷ (N − 1)
When N is odd, the number of intervals is even and the calculator applies Simpson's one-third rule:
AWP ≈ Δx ÷ 3 × [B0 + Bn + 4(B1 + B3 + ... + Bn−1) + 2(B2 + B4 + ... + Bn−2)]
When N is even, the number of intervals is odd. The current calculator then uses the trapezoidal rule as a fallback rather than applying a mixed Simpson rule.
Method selection: use an odd number of stations when a Simpson one-third-rule result is required. An even number of stations produces the calculator's trapezoidal fallback and may give a different result for strongly curved waterplane outlines.
Station values must:
The polygon mode calculates the area of a waterplane outline from ordered coordinate pairs using the shoelace formula:
AWP = 1/2 × |Σ(xiyi+1 − xi+1yi)|
Use x as the longitudinal coordinate and y as the transverse coordinate. Points must follow the waterplane boundary in order. The calculation closes the polygon between the final and first points automatically.
In full-polygon mode, enter the complete waterplane perimeter. In one-side mode, enter one side of a symmetric waterplane from one centreline endpoint to the other. The calculator mirrors the intermediate points about y = 0 before calculating the area.
One-side coordinate requirement: the first and last points should lie on the centreline. The method assumes port-starboard symmetry and is not suitable for an asymmetric waterplane without entering the complete polygon.
Waterplane area determines the vessel's local rate of displacement change with draft and is therefore directly related to TPC.
The distribution of that area also affects the waterplane's transverse and longitudinal second moments of area. These moments enter the calculation of transverse and longitudinal metacentric radii:
BMT = IT ÷ ∇
BML = IL ÷ ∇
CWP gives only an area ratio. It does not provide IT, IL, the longitudinal centre of flotation or the detailed waterplane outline.
Two waterplanes can therefore have the same CWP while producing different initial-stability and trim characteristics because their areas are distributed differently.
A lower value indicates a more tapered waterplane relative to its waterline rectangle. A higher value indicates a fuller waterplane with a greater proportion of its maximum breadth maintained over the length.
The ranges below are broad comparison values rather than design or acceptance limits.
| Waterplane or vessel form | Indicative CWP | General interpretation |
|---|---|---|
| Fine or strongly tapered waterplane | Approximately 0.65–0.80 | Relatively fine ends and limited parallel breadth |
| Container, Ro-Ro and general cargo forms | Approximately 0.78–0.88 | Moderately full merchant-ship waterplane |
| Bulk carriers and tankers | Approximately 0.85–0.95 | Full waterplane with substantial parallel middle region |
| Barge-like or nearly rectangular waterplane | Approximately 0.90–1.00 | Waterplane approaching its surrounding rectangle |
Vessel type alone does not determine CWP. Draft, bow and stern shape, parallel middle-body length, transom immersion and the chosen waterline all affect the value.
CWP belongs to a stated waterline. As draft changes, the waterline may intersect different parts of the bow flare, stern shape and bilge geometry.
A coefficient calculated at design draft should not automatically be applied at ballast draft. The same limitation applies when the vessel has substantial trim.
This calculator uses LWL, BWL and the waterplane area supplied by the user. For an accurately trimmed or heeled condition, use waterplane geometry derived from the actual immersed hull condition.
Waterplane coefficient is useful for describing planform fullness, checking waterplane areas and comparing hulls at stated drafts. It can also provide a quick consistency check between AWP, TPC, LWL and BWL.
CWP alone cannot determine:
With a valid waterplane area and matching waterline dimensions, CWP should be greater than 0 and no greater than 1. A result above 1 normally indicates mismatched dimensions, incorrect units, duplicated breadths or invalid polygon coordinates.
The terminology and hydrostatic relationships used on this page follow established naval architecture references:
NauticalSolver calculators are intended for preliminary engineering, study and independent checking. Use approved hydrostatic particulars and vessel-specific geometry for operational, contractual or statutory work.