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I am attempting to calculate area of country polygons using the field calculator.

The layer in question is unprojected WGS84, but I have set the project CRS to World Equidistant Cylindrical (Sphere), which is listed with the projected CRSs. I've also enabled on-the-fly transformation.

From my limited understanding of the program (I am running QGIS 1.8.0), the measurements should be given in the units of the project CRS, and the proj4 string for the World Equidistant Cylindrical seems to indicate that its units are in meters.

Nonetheless, the measurements are off by orders of magnitude. For example, Afghanistan is significantly larger than 92 square meters, or even 92 sq km. I can only assume I am getting area in degrees. What must I do to change that?

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Your only option is to reproject your data into a CRS that has a linear measurement unit - such as World Equidistant Cylindrical. On-the-fly projection means QGIS is converting degrees to meters only within the context of your workspace - the data itself does not get modified at all. When you use the field calculator, you're performing operations on the original WGS84 dataset, not the on-the-fly transformation of the dataset.

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