The following question was asked in the comments section of the article Archimedes' Principle Explained.
Q: Can you just explain how to measure the volume of an irregularly shaped object which floats in water? Can we use Archimedes' principle? How?
A: It is certainly possible to do so using Archimedes' principle, provided the density of the object is known. The key here is to find the mass of the object. Then we use the mass formula Mass = Vρ (where V is the volume of the object and ρ is the density of the object) to find the volume.
In my article Archimedes' Principle Explained, there is a section about objects floating freely. It is mentioned that
Weight of floating body = Weight of fluid displaced
Mass of floating body = Mass of fluid displaced
So if we can know the mass of water displaced (e.g. measuring the apparent increase in volume of water and multiplying it with its density, as in finding mass using the mass formula) when the object floats in water, we can know the mass of the object.
By rearranging the mass formula and substituting the mass and ρ into it, we can have the volume of the object. However, we must know the value of ρ beforehand.
Note: The above answer does not take into account other ways of finding the volume of an object that floats in water other than using Archimedes' principle.