You cannot flatten an orange peel without tearing or stretching it, and you cannot flatten the Earth without doing the same to distance, area, shape or direction. This is not a limitation of cartography that better software will fix — Gauss proved in 1827 that a sphere and a plane have different curvature and no map between them can preserve everything. So mapmakers choose which error to accept. Below you can see exactly which one each famous map chose, and how big it gets.
Watch the graticule as you switch. On the Mercator the spacing between parallels grows without limit toward the poles — that stretching is not decoration, it is the price of keeping every compass bearing a straight line, which is why sailors wanted it. On Gall–Peters the parallels bunch up instead, because keeping area honest means squashing what Mercator stretched. Robinson refuses to get anything exactly right and instead spreads the error around until the map merely looks correct.
Each of these was a perfect circle of the same size on the globe — 500 km across, drawn on the sphere and then projected, not drawn on the flat map. On Mercator they stay circles (that is what conformal means: shapes survive locally) but swell enormously with latitude. On Gall–Peters every circle encloses exactly the same area, which is the whole point, but near the poles they flatten into lozenges. Robinson does a bit of both. No projection can keep them all the same size and all circular.
Pick a landmass and drag it north. Its real size never changes — it is the same patch of sphere — but the map redraws it larger and larger. Take Greenland down to the equator and it shrinks to something like Algeria, which is roughly the truth. Then put Africa up where Greenland lives and watch it swallow the map. The apparent area below is measured off the projected outline itself; the true area is measured on the sphere.
On a Mercator map the scale at latitude is sec(lat) in every direction, so area is multiplied by sec squared. That function is 1 at the equator, 2 at 45 degrees, 4 at 60, and past that it simply runs away: 10.5 at 72 degrees, where the middle of Greenland sits, and 132 at 85. At the pole itself it is infinite, which is why every Mercator map you have ever seen is cut off somewhere before it — this one stops at 78 degrees. The equal-area line is flat at 1 by construction, and that flatness is bought entirely with shape.
True areas are the published figures. The inflation column is the average of sec squared over each outline, computed by integrating across the shape rather than reading it off at one latitude — a country spanning 25 degrees of latitude does not have a single scale factor. The last column is the size it appears to be. Greenland ends up looking marginally larger than Africa while being one fourteenth of it, and that single row is why the argument about this projection has been running since the 1970s.
| landmass | true area | looks this many times too big | apparent area on Mercator |
|---|
The projection formulas are the real ones: Mercator y = ln tan(pi/4 + lat/2), Gall–Peters as a cylindrical equal-area with standard parallels at 45 degrees, and Robinson as its published table of nineteen control points at five-degree intervals, interpolated linearly between them, with the standard 0.8487 and 1.3523 coefficients, which reproduce the projections documented 1.9716 width-to-height ratio. Area on the sphere is computed with the Chamberlain–Duquette formula, which returns 1.2302 million square kilometres for a ten-by-ten degree box at the equator against an exact 1.2302. The distortion figures are integrals, not guesses: shrink the test patch to a fraction of a degree at 72 degrees north and the measured inflation converges to 10.4722, against sec squared of 72 degrees = 10.4721.
The coastlines are not. They are hand-simplified outlines of twenty to forty points, and they come out between four and thirteen per cent away from the surveyed areas — which is why the table quotes published areas for truth and uses the outlines only for the ratio, where the simplification largely cancels. Everything here also treats the Earth as a perfect sphere; it is an oblate spheroid about 21 km wider at the equator than pole to pole, which real cartography must handle and which shifts areas by a fraction of a per cent. Tissot circles are drawn as true spherical caps rather than the infinitesimal ellipses of the formal theory, so a large one at high latitude shows slightly more exaggeration than the point value at its centre — 11.9 rather than 10.5 for a 1000 km circle at 72 degrees, because its northern edge is at 81 degrees where the stretching is far worse.