The coordinate system used in the Map-Plane Model has the +x-axis
directed along the Prime Meridian (
longitude) and the +y-axis
east of that. A distance of 10,000 km from equator to pole is
used. With this the radius of the Earth is
The number of km per degree of longitude is
With these we use the following transformation from latitude and longitude to X and Y coordinates in km:
This coordinate system works well, and produces only moderate distortion as long as the point is near the pole. In order to quantify the error in calculating area (and hence volumes) in this coordinate system, we must compare areas calculated rigorously on the surface of a sphere with those calculated in this projected coordinate system.
First let us look at areas calculated in the projected coordinate system. For a circle centered on the pole, the area is
where 
is the angle in radians from the pole (
in
radians). Thus the area of the entire Northern Hemisphere
(
, corresponding to the equator) is 
.
For the area calculation on a sphere, we will also use a circle centered at the pole. The area here will be
Thus the actual area of the entire Northern Hemisphere (
,
again corresponding to the equator) is 
. We can see
that the maximum error in areas is an overestimate of the area in the
projected coordinate system of about 23%.
Forming the ratio
we can see that the limit as 
goes to zero will approach one,
indicating little distortion near the poles.
This analysis of the error in calculating areas actually may underestimate the problem, because it includes the total area, much of which is close to the pole where the distortion is small. If we wish to calculate the error as a function of latitude, we want to calculate the areas of annuli of finite width at different latitudes. In the projected coordinate system the area of such an annulus is
where 
and 
are the two latitudes enclosing the annulus.
Similarly for the area of the annulus on the surface of the sphere we have
and the ratio is
Here the error is more extreme at latitudes close the the equator, with distortions of up to 50%.
With all this it is still reasonable to reflect that in areas of
interest at latitudes above 
N, the overall distortion is still
less than 7% and the differential distortion is less than 15%.
Applying this correction to the Antarctic, we only find a 1% overestimate in the volume using the uncorrected projected areas.