Measurement Problems

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I'm working on my first build (1985 Murray Monterey Beach Cruiser) and feel like an idiot so far.I have many questions but i will start with my current problem.I want to change the headset.I bought a 1" headset off Ebay thinking it would fit but it's not right.It's all the same except the new cup is too small for my tube.The new cup measures about 1-3/16" where it goes in the tube and the old cup is 1-3/8".I see when looking for headsets that they are measured ??mm x ??mm x ??mm.I don't know which measurement is which.I guess my question is can i find a headset for this bike online and if so what size would i look for?.I have also seen spacers when looking for headsets.Will i need to use spacers if i can't find the size headset i need or is that even an option for what i'm doing?.
 
I usually 'google' up 'mm to inches' or 'cm to inches' and click on the calculator site, a box comes up to enter either and hit go, theres your answer (you can go the inches to mm/cm too.
The spacers are to get 'stack height' (and avoid cutting it off) I think, though there might be cup 'shims' to make the headtube ID???
You, or a friend with a lathe could turn some for your set OD to the head ID, and they should fit in firmly. I would rough them out for a light press or Locktite fit onto the cups and turn the OD with the assembly mounted on an expanding mandrel to that perfect custom, in the headtube, fit.
Sorry to say, a lot of this stuff seems to be in various incarnations of a standard that is anything but, or only nominal. Also, to go by fractional measurements is only roughly close, not near enough for fine fitting parts.
 
I appreciate the input.Sheldon Browns site did help.I had already converted my measurements but was just unsure how to phrase them correctly to find what i was looking for.I still have had no luck in the search and decided i will need to refurbish what i have or make shim's as suggested.The size headset i came up with for my bike would be 22.2mm x 25.4mm x 34.92mm.The 34.92mm being the 1-3/8" id of my tube and it's the 34.92mm that's killing me, can't find it online.Thanks again for the help and any leads would be appreciated.
 
I've read other posts of issues with this model of Murray having an oddball headset cup size, do a search on "headset murray monterey" I believe others have solved this problem already. If all else fails that's close to a 1 1/8" headset cup size(34mm) and Pyramid makes a bunch of stuff that can be mixed and matched, or possibly use a set of Problem Solver brand headset reducers either may need a slight shim.
 
FWIW, When you are close, but close isnt quite good enough: Ive had several projects that required about .010-.020"ish shimming (if your 'mix & match' gets within that) and used alum. pop can strips for quick & dirty, or annealed automotive feeler gauges for way tough.
Lately I have been trialing heating system 'foil tape'; its a real alum. foil tape on a roll for heavy duty ductwork. Cut the strips, peel, and stick on to fit up. I dont know if this will help your situation, but it has let me fit some problematic bits and at least get final fit dimensions. Because it is a tape and sticks really well it is way easier to use than the alum. pop can strips.
 
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