luca76 February 2016
You could use the field calculator. See this help page:
All you do is to open the Attribute Table of the shapefile, from there enable the editing mode, open the calculator, add a field and use the $area variable in the expression list.
Addendum: You have to split one multipolygon to many polygons.
Do do it, go to menu
Vector -> Geometry Tools -> Multipart to singleparts
with it, you create a new shapefile with many polygons inside. Then you can calculate the area in the attribute table, as said before.