Rule of thumb · EnvironmentalNº 02 / 186

Mercator squares the stretch

Why does Greenland look as big as Africa?

On a Mercator map, area is exaggerated by 1/cos²(latitude) — four times at 60°, fifteen times at 75°.

Why it works

Mercator keeps angles true, and to do that it must stretch east–west and north–south by the same factor, 1/cos(latitude). Area is the product of the two, so the error is that factor squared. Greenland spans 60° to 83°, so Mercator draws it larger than Africa — which is really about fourteen times its size.

When it fails

It is not a flaw, it is the trade. A conformal map cannot also be equal-area — Gauss proved in 1827 that no flat map of a curved surface preserves both — and preserving angle is exactly what makes a constant compass bearing a straight line, which is why Mercator has never been replaced for navigation. The rule also breaks down at the poles, where cos(latitude) goes to zero and the projection runs to infinity; every Mercator map is clipped somewhere around 85°. And Greenland’s own area is quoted two ways: about 2.17 million km² including the ice sheet, and 410,000 km² of ice-free land — the second is what the World Bank reports and what the atlas therefore shows.

Do it exactly

Estimate with the rule, then check it against the calculator that models it properly.

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