Manually updating the attribute table of a clipped shapefile

I need to do calculations of surfaces of polygons in vector layers. The clip tool does just what I want – I can clip the polygons of the input layer to the clipping layer. When I project the new layer I can see that the polygons are indeed clipped along the polygon border of the clipping layer. So that worked.

However, when I check the attribute table, the surface of the clipped new polygon is still equal to the original polygon surface. In the manual it says: "The attributes of the features are not modified, although properties such as area or length of the features will be modified by the clipping operation. If such properties are stored as attributes, those attributes will have to be manually updated."

Now, HOW do I manually update these attributes? All instructions I have found so far stop after the clipping and none goes into the updating of the attribute table…