section / cut
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section / piece
A separated part, a section, or an individual piece of a larger whole. It can refer to routes, areas, materials, or objects.
Das letzte Teilstück des Wanderweges führt steil bergab.
The last section of the hiking trail leads steeply downhill.
The word is made up of 'Teil' (part) and 'Stück' (piece). So it describes a piece that is part of something larger. The 's' in the middle is a linking sound that connects the two words.
Die Bauarbeiten betreffen nur ein kurzes Teilstück der Straße.
The construction work only affects a short section of the road.
To say what something is a section of, you usually use the genitive or a 'von' (of/from) construction. For example: 'das Teilstück des Rohres' (the section of the pipe) or 'ein Teilstück von der alten Mauer' (a section of the old wall).
Ein Teilstück der Brücke muss erneuert werden.
A section of the bridge has to be replaced.
When 'Teilstück' (cut/section) is used in the context of cooking, eating, or a butcher shop, it almost always means a specific cut of meat.
Der Metzger empfahl mir dieses Teilstück vom Schwein für den Braten.
The butcher recommended this cut of pork to me for the roast.
part
'der Teil' and 'das Teilstück' both mean something that belongs to a larger whole. 'der Teil' is more general and can be used for almost any section or piece. 'das Teilstück' is more specific: it usually means a clearly defined or separated piece, for example of a route, an area, a material, or an object. The words can be confused because a 'Teilstück' is always also a 'Teil'. But not every 'Teil' is automatically a 'Teilstück'. 'Teilstück' sounds more appropriate when you mean one individual, specific piece of a larger whole.
Amira markiert das Teilstück des Weges, das durch den Wald führt.
Amira marks the section of the path that leads through the forest.
Welcher Teil der Aufgabe war für dich am schwierigsten?
Which part of the task was the most difficult for you?