Thank you. Got it now. I've heard builders refer to the roof as in getting the lid on, but never hat. I certainly would never have worked this one out, even though I guessed the answer.
Don't ask me why but this discussion of hat = tile and finding myself lost reminds me of the French. Toit meant roof and trottoir meant pavement - I could never get these two words the right way round and used to end up in a right mess. Isn't language wonderful?