der Dorn
thorn
A sharp, prickly growth on a plant, such as on roses, cacti, or thistles.
Vorsicht, ich habe mich am Dorn einer Rose gestochen.
Careful, I pricked myself on the thorn of a rose.

thorn / awl
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thorn
A sharp, prickly growth on a plant, such as on roses, cacti, or thistles.
Vorsicht, ich habe mich am Dorn einer Rose gestochen.
Careful, I pricked myself on the thorn of a rose.

The meaning of 'Dorn' becomes clear from the context. If people are talking about plants such as roses or cacti, it means the sharp thorn. In connection with tools or crafts, it means a pointed instrument. The fixed expression 'im Auge' (in the eye) always points to the idiom.
Der Kaktus hat sehr lange, gefährliche Dornen.
The cactus has very long, dangerous thorns.
In connection with plants, people often speak of 'spitze Dornen' (sharp thorns). The typical verb is 'sich stechen an' (to prick oneself on) with the dative, e.g. 'sich an einem Dorn stechen' (to prick oneself on a thorn).
Pass auf, dass du dich nicht an den Dornen der Brombeerhecke stichst.
Be careful not to prick yourself on the thorns of the blackberry hedge.
The idiom 'jemandem ein Dorn im Auge sein' (to be a thorn in someone's side) has a fixed structure. The person who is annoyed is always in the dative.
Die neue Autobahn ist den Anwohnern ein Dorn im Auge.
The new highway is a thorn in the residents' side.
The meaning 'spitzes Werkzeug' (pointed tool) is used almost exclusively in technical or craft contexts. In everyday life, this meaning is rare.
Mit einem Dorn wurde das Leder vorgelocht.
The leather was pre-punched with an awl.
tip / top / lace
'die Spitze' and 'der Dorn' both have to do with something pointed. 'die Spitze' is the narrow end of an object and is a very general word. 'der Dorn' is usually a pointed part on a plant or a pointed tool part. A thorn can prick, but not every tip is a thorn. Learners may confuse the words because both are pointed. With plants like roses, you usually say 'Dorn'; with pencils, knives, or mountains, 'Spitze' is more likely.

Beim Schneiden der Rosen hat sich Anna an einem Dorn verletzt.
While cutting the roses, Anna injured herself on a thorn.
Die Spitze des Bleistifts ist stumpf, deshalb spitzt Tom ihn noch einmal an.
The tip of the pencil is blunt, so Tom sharpens it again.