¿ How much does a ship weigh ?

Archimedes' principle

This question is deliberately ambiguous and what intends to ask is how can you get the weight of a ship, referring, of course, to a large ship, that can not be placed on a balance. It also refers to the weight of a real floating ship  already built and it is not considering here the weight estimation during the ship's design phase.

The answer to the question is on the second line, Archimedes' Principle (Ref. (1)), that, we recall, says:

A solid body immersed in a fluid experiences a vertical upward force equal to the weight of the fluid volume it displaces, which is called displacement.

 Annex 1 explains its proof.

If a ship is freely floating at rest it is because her weight is exactly equal to her displacement. If the weight would be greater than the displacement, the vessel would immerge some amount increasing its displacement to achieve the balance at a greater draft. If balance is not achieved  the ship will sink up to the bottom. Alternatively, if the weight is less than the ship displacement,  she would emerge a certain amount to reach equilibrium at a lower draft.

Hence it is clear that if we measure the ship displacement that same number will be her weight.

In general, the weight of a floating vessel is made up of the weight of vessel itself(hull, machinery, equipment, accommodations), plus the weight of liquids in tanks (fuel, oils, fresh water, ballast water etc.), weight of food, stores, baggage, people on board, and the weight of cargo in her holds or tanks.

Measuring the displacement is a simple office task. But you need a basic information, which is the drawing which represents the ship's hull form. If the hull would of a simple geometric shape, like a parallelepiped, a cylinder, a sphere and so on, the displacement should be calculated by known geometry formulas. In the simplest case of a box shape hull, the displaced volume is the product of the length by the breadth (width) by the draft (height). The length and breadth are constants that are known and shown in the basic ship drawings. The draft is variable and must be measured on the ship, reading where the water surface gets on the draft scales  marked on the hull, fore and aft (in large vessels also in one or more intermediate positions).

Finally, multiplying the above volume by the water density (in Tm/m3) you get the displacement in tonnes, if the dimensions are in meters.

Nevertheless most ships do not have box forms, but others much more complicated, due to hydrodynamic reasons, and do not generally correspond to known geometric shapes to which it could be applied formulas to calculate their volume.

This figure shows a typical hull forms of merchant ships, which normally are drawn the in 3 projections,the so-called lines plan, being the transverse the most interesting one to calculate the displacement. This projection is called body plan (see figure). Along the length are drawn a number of cross sections, typically 21, numbered from 0 (aft) to 20 (fore).

From the drafts read at fore and aft marks (which generally are not the same) the draft at each of the 21 cross sections is calculated and the sectional areas until these drafts are evaluated, and then a numerically integration of all the areas is done to get the displaced volume. All this calculation can be done manually, but usually with a computer program.

In the vessels' every day life it is not necessary to make so many calculations, because the shipyard  delivers drawings and tables already calculated to determine the displacement corresponding to the any drafts read on the mark scales.

Once you know the full ship weight, if you substract her own light-weight (calculated by the shipyard after completing the construction), the weights of liquids on board (as defined by the soundings fitted in each tank), and weights of food, supplies and people (as defined by their records), we obtain the weight of the payload stowed in the holds or tanks, a data very useful and necessary for the ship operation.

Appendix 1 - Archimedes' principle proof

The force exerted on the floating body by the liquid is the result of the liquid pressure on each portion of the submerged surface. If we imagine a submerged surface identical to that of the body in question, virtually separated from the rest of the liquid, it is clear that the liquid contained within this virtual space will be in equilibrium, so that will be equal its weight and the force  received from the surrounding liquid. As a result, as the two forces are equal (because acting on the same form) so their weights must also be the same.

References
Ref (1) - http://en.wikipedia.org/wiki/File:Archimedes_pi.svg