I have this shapefile containing what looks like polygons but the attribute table only contain a "level" column.
How would I fine the area of each of the polygons and add that to the attribute table?
Example files: https://www.dropbox.com/sh/mgcppedd5kvb268/AABmFCXaG39IEV4hV_q2QchPa?dl=0
You could use the field calculator. See this help page:
http://docs.qgis.org/2.8/en/docs/user_manual/working_with_vector/field_calculator.html
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.