Can you use "a piece of" and "a part of" interchangeably?
Obviously there are exceptions: A piece of cake, a piece of pizza, a piece of any circular object.
I personally don't seem to be able to find any situation where they're perfectly interchangeable, so I'd say no. To me, "a piece" seems smaller than "a part", and a "piece" sounds like it's severed/to be severed from the whole, while a "part" can still be/remain part of the whole. (Your examples, or e.g. "a piece of advice", which means it's a small advice.) You can say "I disagree with a part of it" or "a part of me feels...", and these don't mean the part is to be literally cut off, or that they're small.
I've never bothered learning these things above to be able to use the terms, instead, I've developed an instinct telling me which to use. So, "just wondering" is right!
p.s. I'm not a native.
(Minor edit to correct a mistake)
In an abstract sense, yes, but not with concrete nouns. You could say 'I've lost a piece of myself' or 'a part of myself' but you couldn't say 'a part of art' or 'I'll have another part of pie'.
If I had to come up with a very loose rule for it, I'd say that you use 'piece' to describe bits of something which is a single, inseparable object and either 'part' or 'piece' for anything which can be taken apart, like a car.
Like everything in English, there are bound to be numerous exceptions. :)
Most of the time, a 'part of' something is usually seen as a percentage, or 'unknown' portion. (When I use it anyway). When something is a 'piece of' it's generally a small, identifiable object, which fell off, or is a part of a larger whole. E.g.,
A piece of the puzzle/cake (object which can be divided) is missing!
A part (portion) of the painting is smudged.
A part of my car fell off, and now my car doesn't work... (I don't know what fell off...)
A piece of my car fell off, I don't know where it fits... (I'm holding it in my hand/have it on my shelf; not a clue how to go about fitting it!)
Part of the puzzle is missing - an area which can consist of multiple pieces as the exact amount isn't known.
Summary: Part(s), Portion(s), Area(s) and Segment(s)... | ...Piece(s), Fragment(s), (Broken Stuff)
Generally yes but then there are exceptions as you have pointed out, One can say "a piece of my mind .....but not a part of my mind. or interchangeably we are going to have a piece of that land or ....a part of that land. It doesn't necessarily have to be a circular piece or part of a car/ table equipment etc. As an english speaker I thought this was a really interesting question thanks
Interesting question, given that the first definition of "part" in The American Heritage Dictionary makes it synonymous with "piece". I think the real issue is that each of those words is used often in idiomatic expressions in which the two cannot be interchanged and mean the same thing (e.g., a piece of art, a piece of my mind, a piece of the action, "I am a part of all that I have met", playing a part, playing a piece, and so on.) Perhaps we just have to play it by ear.