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I'm working with QGIS 3.10, using the diagrams function for vector layers. Is it possible to label each single diagram sector with an attribute / calculated percentage?

I read this article: Label QGIS diagram sector with percentage etc, but it´s 4 years old.

Is there another way to achieve this today (QGIS version 3.10 or 3.12)? Especially directly in the QGIS.

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Yes it is possible but not very sophisticating: you can use a rule based labeling for every category without any condition in your rule and then set the option to show all labels including colliding labels for all of your rules, see pictures below...

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