The matter is quite complex but not complicate. Better you go to use Excel to calculate. Then it’s your decision what you need to put into the calculation. The most common method uses to split up the shape by triangles and then… The mathematical formulas are not so heavy as you would like to think. But it takes time doing that for a very complex shape.
I do know how to calculate centre of area by triangles and trapeziums – but it simply takes too long to do it for hundreds of different components that are all very different shapes. For many items. it's often much quicker to print the shape out on stiff paper, cut it out and balance it on a knife edge ... I know that's not a very hi-tec solution
MaxSurf (my naval architecture program) calculates the centres of area of all the separate 3D curved and 2D flat surfaces that I create in it. Before CAD I used to have to do this by hand, using numerical calculus (Simpsons multipliers), and it was very time consuming. Then along came Appleworks, and at least I could speed up the process. Now the MaxSurf does it (still using Simpsons) and it is in practical terms instantaneous. Which is what CAD is all about really – speed and accuracy. But I can't effectively put all the components that go into a boat into MaxSurf.
RealCADD does a great job of telling me the area of a shape like this for example, by me subtracting the area of the apertures from the basic outline (all of which are closed polygons):
But it would speed things up considerably if it could also calculate the centre of area! Just for the separate closed polygons – not for the whole component. I can then calculate the centre of the composite component by moments on a spreadsheet, as I do for the whole boat.
I appreciate that RealCADD is not the price of say Catia 5 (which is probably what I would use in a perfect world), but airplanes sell for millions and small boat designs only for hundreds!
So I do what many designers do – work out the centre of gravity of typical components and take an educated guess at the rest. And on top of that, manufacturers are horrible – very rarely giving you the centre of gravity of things like engines, winches, air conditioning units and so on. So the whole thing becomes a bit of an educated guess – and very tedious as well.
But people set out to sea in what I design and it behooves me to make as good a job as I possibly can to make the designs stable and safe – and knowing the centre of gravity of the whole boat is central to that!
PS I don't know that Rag Time – the only Ragtime I recall was designed by John Spencer (a New Zealander) – but I think she was originally called Infidel. She is still going strong, winning long distance races.