It's easy to figure out a ship's weight and the longitudinal (LCG) and transverse (TCG) centroids from just observing its drafts, trim and heel. Weight as function of the volume of water of water displaced...Archimedes figured that out it two thousand years ago (Hey, Eureka!). But determining the vertical centroid, the VCG, is different and has to be determined by an inclining experiment. Here's a pretty good explanation of the process at the naval post graduate school website.
The elements of the experiment are moving precisely measured weights, precise distances transversely in the ship to cause heel, then measuring (precisely) that heel angle. This is the relationship between the heel and weight moment (weight x distance).
The weight or displacement of the ship, , is known. The transverse moments and resulting heel angles are measured. Solve for GM. From GM, and the ship hydrostatics, derive the VCG.
In theory easy, in practice...precision is everything. The standards and precision for conducting the test are in our federal regulations. A vessel must be heeled to a maximum angle between two and four degrees, the angle to be measured precisely. We commonly use a pendulums or water filled clinometers. Here's what they look like:
I say pendulums plural because three separate devices are required (and I think insisting on "pendula" is affectacious). To get the required precision, the minimum greatest deflection is six inches. To get that deflection for two degrees of heel, the pendulum length has to be 172" (4.4m). A pendulum that long on a boat on a windy day wiggles around quite a bit...it can be hard to get an accurate reading.
We're conducting the inclining experiment on the Alcatraz Flyer on April 11th. It's a very wide and stiff little ship, and to get the required minimum of two degrees of heel, we have to move thirteen tons of weight in a specified set of movements across the deck. It takes up most of the day.