der Ärmel
sleeve
The part of a piece of clothing that covers the arm.
Ich habe mir Tomatensoße auf den Ärmel gekleckert.
I spilled tomato sauce on my sleeve.

sleeve
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sleeve
The part of a piece of clothing that covers the arm.
Ich habe mir Tomatensoße auf den Ärmel gekleckert.
I spilled tomato sauce on my sleeve.

The word 'Ärmel' (sleeve) is closely related to 'der Arm' (arm). It refers to the part of a piece of clothing that covers the arm.
Dieses T-Shirt hat sehr kurze Ärmel.
This T-shirt has very short sleeves.
The idiom 'die Ärmel hochkrempeln' does not only mean literally to roll up one's sleeves. It is often used figuratively and then means 'to prepare for hard work' or 'to get stuck in energetically'.
Okay, lasst uns die Ärmel hochkrempeln und mit der Gartenarbeit anfangen.
Okay, let's roll up our sleeves and start the gardening.
The idiom 'etwas aus dem Ärmel schütteln' has a fixed structure. It describes the ability to deliver a task or an answer seemingly without effort and spontaneously.
Er kann komplizierte Rechnungen einfach so aus dem Ärmel schütteln.
He can pull complicated calculations out of his sleeve just like that.
arm
'der Arm' and 'der Ärmel' are closely connected, but they do not mean the same thing. 'der Arm' is a body part, from the shoulder to the hand. 'der Ärmel' is the part of a piece of clothing that covers the arm. Learners may confuse the words because both often appear in the same situation, for example when getting dressed or when clothing does not fit. Memory tip: The arm belongs to the body; the sleeve belongs to clothing.

Beim Kochen ist der Ärmel meiner Jacke nass geworden.
While I was cooking, the sleeve of my jacket got wet.
Mein Bruder kann den Arm heute kaum bewegen, weil er gestürzt ist.
My brother can hardly move his arm today because he fell.