The area of the section obtained by cutting the hull at the water surface is called the waterplane area, and the centroid of that plane is called the center of flotation.

The waterplane area also indicates how much the displacement volume increases or decreases as the draft changes. This article explains how to calculate the waterplane area and the center of flotation, and relates them to tonnes per centimetre immersion and to the change in draft when the density of the water changes.

1.5.1 Calculation of Waterplane Area

Let $x[\mathrm{m}]$ be the distance measured forward from the aft perpendicular (A.P.), and $z[\mathrm{m}]$ the height measured vertically upward from the baseline (B.L.). Let $b(x,z)[\mathrm{m}]$ be the half-breadth at that position, measured from the hull centerline to the side of the hull on one side.

Here, we consider a hull that is symmetric to port and starboard, with no heel and with the baseline parallel to the water surface. Let $d[\mathrm{m}]$ be the draft, and $L_{PP}[\mathrm{m}]$ the length between perpendiculars, the horizontal distance from the aft perpendicular (A.P.) to the forward perpendicular (F.P.).

If the horizontal section at height $z$ is divided along the length into small widths $dx[\mathrm{m}]$, the area of each portion is $2b(x,z)\,dx[\mathrm{m^2}]$. Summing these along the length of the ship gives the horizontal sectional area $A_W(z)[\mathrm{m^2}]$.

The section at the height of the water surface $z=d$ is called the waterplane, and its area $A_W(d)$ is called the waterplane area.

1.5.2 Calculation of the Center of Flotation

The centroid of the waterplane is called the center of flotation and is denoted by the symbol $F$.

Let $y[\mathrm{m}]$ be the transverse coordinate measured from the hull centerline, and let $(x_F,y_F)$ be the coordinates of the center of flotation.

The longitudinal position of the center of flotation $x_F(d)[\mathrm{m}]$ is obtained by multiplying each small area of the waterplane by its distance from the A.P., summing the results, and dividing by the waterplane area. The numerator is the first moment of area about the transverse axis through the A.P.

For a waterplane that is symmetric to port and starboard, the center of flotation lies on the hull centerline (C.L.), and $y_F=0$.

When the position of the center of flotation is expressed relative to midship $M$, taking the aft direction as positive, $\overline{MF}=\frac{L_{PP}}{2}-x_F$. If this value is positive, the center of flotation is aft of midship; if it is negative, it is forward of midship.

The center of flotation always lies within the waterplane, even when the hull rolls or pitches.

1.5.3 Relationship Between Waterplane Area and Displacement

If the hull is divided vertically into small thicknesses $dz[\mathrm{m}]$, the volume of each portion is $A_W(z)\,dz[\mathrm{m^3}]$. Stacking these from the baseline up to the water surface gives the displacement volume $\nabla(d)[\mathrm{m^3}]$.

From here on, displacement is treated as the mass of the water displaced and is denoted by $\Delta(d)[\mathrm{t}]$. If the density of seawater is $\rho_{sw}[\mathrm{t/m^3}]$ and is taken as constant, the displacement is given by the following equation. The subscript sw stands for seawater.

To examine how much the displacement increases when the draft increases slightly, Eq. (1.5-4) is differentiated with respect to the draft $d$, giving the following equation.

The unit of the left-hand side is $\mathrm{t/m}$, and it represents the rate of change of displacement with respect to draft. When floating in water of the same density, the larger the waterplane area, the larger the increase in displacement for the same small increase in draft.

1.5.4 Tonnes per Centimetre Immersion (TPC)

Tonnes per centimetre immersion (TPC) represents the increase or decrease in displacement corresponding to a change in draft of $1[\mathrm{cm}]$. Its unit is $\mathrm{t/cm}$.

If the waterplane area can be regarded as nearly constant over the change in draft, then, since $1[\mathrm{cm}]=0.01[\mathrm{m}]$, Eq. (1.5-5) can simply be converted to a value per centimetre.

When using numerical values with the density expressed in $\mathrm{t/m^3}$ and the waterplane area in $\mathrm{m^2}$, dividing their product by 100 gives the numerical value of TPC in $\mathrm{t/cm}$.

When a ship sinks bodily without inclining, TPC also corresponds to the mass that must be added to increase the draft by about $1[\mathrm{cm}]$. For a ship whose waterplane area varies with draft, TPC also differs for each draft.

1.5.5 Change in Draft When Moving from Seawater to Fresh Water

Consider the case in which the same ship moves from seawater to fresh water without any change in its load. For a ship floating at rest, the mass of the water displaced equals the mass of the ship, so the displacement $\Delta[\mathrm{t}]$ does not change.

However, because fresh water has a lower density than seawater, a larger displacement volume is needed to displace the same mass of water.

Let the densities of seawater and fresh water be $\rho_{sw}[\mathrm{t/m^3}]$ and $\rho_{fw}[\mathrm{t/m^3}]$, respectively. The subscript fw stands for fresh water.

Letting the displacement volume in seawater be $\nabla_{sw}[\mathrm{m^3}]$ and that in fresh water be $\nabla_{fw}[\mathrm{m^3}]$, the difference between them is as follows.

Let the drafts in seawater and fresh water be $d_{sw}[\mathrm{m}]$ and $d_{fw}[\mathrm{m}]$, respectively. Ignoring any change in attitude and regarding the waterplane area $A_W$ as nearly constant over the change in draft, the increase in displacement volume can be approximated by the product of the waterplane area and the increase in draft.

If $\Delta$ is expressed in $\mathrm{t}$, the densities in $\mathrm{t/m^3}$, and the waterplane area in $\mathrm{m^2}$, the difference in draft given by this equation is obtained in $\mathrm{m}$.

Furthermore, using the TPC in seawater, the increase in draft $\delta d=d_{fw}-d_{sw}$ expressed in $\mathrm{cm}$ is given by the following equation.

Here, $\Delta$ is expressed in $\mathrm{t}$ and TPC in $\mathrm{t/cm}$. Since the density ratio is dimensionless, the unit of the right-hand side is $\mathrm{cm}$.

For example, if the density of seawater is $1.025[\mathrm{t/m^3}]$ and that of fresh water is $1.000[\mathrm{t/m^3}]$, the ratio of the density difference to the density of fresh water is 0.025. Under this condition, 0.025 times the displacement divided by TPC gives the approximate increase in draft in centimetres.

1.5.6 Numerical Calculation Using Hull Geometry

For calculations that include the inclination of the hull, methods have been studied that determine the waterplane at a given attitude from the intersection of the hull geometry with the water surface. In these methods, the hull is represented as a three-dimensional surface, its intersections with the water surface are found according to draft, heel, and trim, and quantities such as the waterplane area and the longitudinal position of the center of flotation are calculated.

Methods that adjust the arrangement of sections according to changes in hull shape, so that the shape can be reproduced with fewer sections, have also been studied. Comparative calculations have shown examples in which the number of sections required can be reduced while maintaining calculation accuracy, compared with arranging sections at equal intervals.

About This Article

References

• An Introduction to Naval Architecture in One Volume, edited by the Maritime College Career Education Research Association
• “A Quasi-Bonjean Method for Computing Performance Elements of Ships Under Arbitrary Attitudes,” Kaige Zhu, Jiao Liu, and Yuanqiang Zhang

Sources

• Transport Canada, “TP 14609 – Study Guide for Applicants to the Fourth-class Engineer Certificate with STCW Endorsement”

(Last accessed: October 18, 2026)


*This article was compiled based on the author’s understanding and practical knowledge, with reference to the above literature.