I have a dataset with polygons of administrative areas for the whole world (https://www.gadm.org/download_world.html). I want to have the area size of each polygon in square kilometers. I saved the GADM file with the CRS "EPSG:54032 - World_Azimuthal_Equidistant - Projected" and used the $area function in the Field Calculator so far. Doing it with "Vector -> Geometry Tools -> Add Geometry Attributes" results in the same outcomes. Even though the results seem to be relatively close to actual area sizes for some units (when comparing them e.g. with area size information from google or Wikipedia), they deviate alot for others.
Am I using the wrong projection or is there another way to calculate area size correctly?
I am using QGIS 3.4.