das Stück
piece
A single part of a larger whole or quantity.
Möchtest du auch ein Stück von meinem Kuchen probieren?
Would you like to try a piece of my cake too?

piece
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piece
A single part of a larger whole or quantity.
Möchtest du auch ein Stück von meinem Kuchen probieren?
Would you like to try a piece of my cake too?

'Stück' (piece) is often used as a counting word for uncountable things or individual objects. In this function, 'Stück' often remains singular even with numbers greater than one: 'zwei Stück Kuchen' (two pieces of cake), 'drei Stück Obst' (three pieces of fruit).
Ich nehme bitte zwei Stück von diesem Käse.
I’ll take two pieces of this cheese, please.
The meanings 'Theaterstück' (play) and 'Musikstück' (piece of music) are very similar. The context makes the difference clear: verbs like 'aufführen' (to perform), 'inszenieren' (to stage), or nouns like 'Bühne' (stage) and 'Theater' (theater) point to a play. 'Spielen' (to play), 'komponieren' (to compose), 'hören' (to hear/listen to), or instrument names point to a piece of music.
Das Orchester spielte ein unbekanntes Stück von Beethoven.
The orchestra played an unknown piece by Beethoven.
When 'Stück' (piece) refers to a part of something uncountable, the noun often follows directly without an article. You say 'ein Stück Kuchen' (a piece of cake), not 'ein Stück ein Kuchen'.
Kann ich bitte ein Stück Papier haben?
Can I have a piece of paper, please?
In a commercial context, 'Stück' (item/unit) is used to indicate the price for a single item. Typical formulations are 'pro Stück' (per item) or 'das Stück' (each).
Die Äpfel kosten 50 Cent pro Stück.
The apples cost 50 cents each.
When 'Stück' (stretch) describes part of a route, it is often combined with adjectives like 'gut' (good/considerable), 'ganz' (whole/quite), or 'weit' (far). It expresses that the distance is considerable.
Von hier ist es noch ein gutes Stück bis zum Bahnhof.
From here, it is still quite a stretch to the train station.
The expression 'am Stück' (in one stretch) means 'without a break' or 'uninterruptedly'. It is often used to describe the duration of an activity.
Er kann einen ganzen Film am Stück ansehen, ohne auf sein Handy zu schauen.
He can watch an entire film in one stretch without looking at his phone.
The idiom 'Stück für Stück' (piece by piece) means 'gradually' or 'step by step'. It describes a process that takes place slowly and in stages.
Er lernt die neue Software Stück für Stück.
He is learning the new software piece by piece.
part
'der Teil' and 'das Stück' can both mean something that belongs to a larger whole. 'der Teil' puts more emphasis on the relationship to the whole or the function within the whole, for example a part of a plan or a device. 'das Stück' often sounds more concrete: you think of something separated off that you can touch, eat, or count, for example 'ein Stück Kuchen' (a piece of cake). The words can be confused because both mean 'not the whole thing'. As a simple guide: for abstract sections or functions, 'Teil' often fits; for concrete, individual quantities, 'Stück' is more likely.

Nach dem Essen nimmt sich Max noch ein kleines Stück Kuchen.
After the meal, Max takes another small piece of cake.
Der letzte Teil der Aufgabe ist für die Prüfung besonders wichtig.
The last part of the task is especially important for the exam.