der Zylinder
cylinder
A geometric solid with two parallel, circular bases and a straight, curved side surface.
Im Mathematikunterricht berechneten wir das Volumen eines Zylinders.
In math class, we calculated the volume of a cylinder.

cylinder / top hat
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cylinder
A geometric solid with two parallel, circular bases and a straight, curved side surface.
Im Mathematikunterricht berechneten wir das Volumen eines Zylinders.
In math class, we calculated the volume of a cylinder.

The meaning of 'Zylinder' is almost always clear from context. In mathematics it refers to the shape, with cars to an engine part, and with clothing or magicians to the hat.
Ein typischer Motor hat vier, sechs oder acht Zylinder.
A typical engine has four, six, or eight cylinders.
In a technical context, 'Zylinder' is often combined with a number to describe the type of engine. For example, people speak of a 'Vierzylinder' (four-cylinder engine) or an 'Achtzylinder-Motor' (eight-cylinder engine).
Mein neues Auto ist ein Sechszylinder.
My new car is a six-cylinder.
The idiom 'ein Kaninchen aus dem Zylinder zaubern' (to conjure a rabbit out of a top hat) means to present a sudden and unexpected solution to a problem.
Wir brauchen dringend eine neue Idee. Kann niemand ein Kaninchen aus dem Zylinder zaubern?
We urgently need a new idea. Can't anyone pull a rabbit out of a hat?
hat
'der Hut' and 'der Zylinder' can both mean a head covering. 'Hut' is the general term for many kinds of hats. 'Zylinder' is only one specific kind of hat: tall, stiff, and cylindrical in shape. The words can be confused because a 'Zylinder' is also a hat. If the exact shape is important, you say 'Zylinder' (top hat). If the type is not important or you are talking about an ordinary hat, you simply say 'Hut'.

Für das Theaterstück trägt der Schauspieler einen schwarzen Zylinder.
For the play, the actor wears a black top hat.
Nach dem Spaziergang legt Herr Schneider seinen Hut in den Flur.
After the walk, Mr. Schneider puts his hat in the hallway.