I added this TextSymbolizer to your giant_polygon style:
<Label><Function name="area"> <PropertyName>the_geom</PropertyName></Function></Label>
Pretty simple. One would guess it displays the area of the styled geometry. I know area computation on geography coordinate systems could produce unusable results.
But whatever coordinate system I use or in which style I implement this the computed value changes with each zoom and it changes slightly with each pan. So what is calculated here? It seems to me it is an scale denominator dependent value. Maybe the pixel area???
If this is the intended behaviour, so how could I get the geometric area of my geometries?
I tested this with GeoServer 2.3.2, 2.4.0, 2.4.4.