Mercator draws Greenland larger than Africa
Africa covers 13.7 times Greenland’s area, yet on Mercator’s map Greenland comes out 8% larger: the map swells everything toward the poles. Carried to the Equator unchanged, Greenland draws at 0.061 of its size at home. Below, the same world in Equal Earth, with ellipses showing what each map does to a circle.
Areal scale: drawn area over true area
- under 0.25
- 0.25–0.5
- 0.5–0.67
- 0.67–0.95
- true, within 5%
- 1.05–1.5
- 1.5–2
- 2–4
- over 4
Controls
Keeps: Area, with continents shaped close to a globe’s.
How It Is Measured
Ten projections, each checked against a published value
Each projection is written out from its published formula for a sphere and checked against a worked example: Snyder’s Map Projections: A Working Manual (USGS Professional Paper 1395, 1987) for eight of them, Snyder and Voxland’s An Album of Map Projections (Professional Paper 1453, 1989) for Robinson’s table, and the PROJ library’s test values for Equal Earth, whose paper is not open to read.
The Ellipses
“An infinitely small circle on the Earth projects as an infinitely small, but perfect, ellipse on any map projection” (Snyder, p. 20). At each point the page differentiates the projection’s own forward formulas, by five-point central differences with a step of 10−5 radian, and takes the scale along the meridian h, along the parallel k, the areal scale s and the largest angular error ω from Snyder’s equations (4-10) to (4-15) and (4-1a). The ellipse on the map is drawn with that same matrix, so the ellipse under the pointer and the numbers beside it are one computation.
The ellipses are drawn far larger than Tissot’s infinitesimal circles, so they can be seen. Each shows the map’s scale at its centre, not the stretching across its own width.
The Areas
Areas are spherical. Each outline is summed as spherical triangles on a sphere of radius 6,371.0072 km, the sphere with the surface area of the GRS 80 ellipsoid (Moritz, 2000). The sums agree with the d3-geo library’s for all 177 countries to 10−12 steradian, and with L’Huilier’s theorem on 200 random triangles.
The Copy
A carried country is turned about the Earth’s centre, along the great circle from its centroid to the point chosen, so every distance and angle within it, and its area, stays as it was. Its drawn size is the area of its outline on the map, copy over home. Beside it, s at the two centres: the areal scale at one point, which is why it differs from the outline’s figure for a country as tall as Greenland.
Cutting outlines at a map’s edge, clipping to a hemisphere or frame, and bending long edges are done by d3-geo. The tests check that d3-geo puts every point where the page’s own formulas put it, to a millionth of a pixel.
The Receipts
| Projection | Checked against | Worst miss | Of tolerance |
|---|---|---|---|
| Mercator | Snyder pp. 266–267: x, y, h, k, inverse | 2.0 × 10−8 | 20% |
| Transverse Mercator | Snyder pp. 268–269: x, y, k, inverse | 4.0 × 10−8 ° | 40% |
| Cylindrical equal-area, Gall–Peters’ family | Snyder p. 278, standard parallel 30° | 2.1 × 10−8 | 21% |
| Albers Conic | Snyder pp. 291–292: x, y, h, k, ω, inverse | 4.9 × 10−8 | 49% |
| Orthographic | Snyder p. 311: x, y, s | 3.3 × 10−8 | 33% |
| Lambert Azimuthal | Snyder p. 332: x, y, the larger scale | 2.0 × 10−8 | 20% |
| Mollweide | Snyder p. 367: θ, x, y, inverse | 1.7 × 10−8 ° | 17% |
| Equal Earth | PROJ more_builtins.gie: metres on a radius of 6,378,137 m to 1 cm, and one inverse to 0.0005° | 2.6 × 10−4 ° | 52% |
| Robinson | Professional Paper 1453, pp. 82 and 223: the table, true scale near 38°, the pole line and central meridian | 0.0028 | 56%, of 0.005 |
| Plate Carrée | Snyder eqs. (12-3), (12-4): h = 1, k = sec φ | 1.6 × 10−11 | under 2% |
| The sphere’s radius | Moritz (2000): GRS 80 authalic radius | 0.12 mm | 23% |
And everywhere, not only at the worked examples: on a grid of 255 points from 80° S to 80° N, wherever each map draws them, every equal-area projection keeps s = 1 to within 10−8, Mercator and Transverse Mercator keep ω under 10−7 radian, and every inverse brings its point back to within 10−9 radian.
One printed figure is wrong, and the test names it. In Snyder’s Transverse Mercator inverse (p. 269), the arctangent of 0.0261859 is printed as 1.4999961°; it is 1.4999988°, so his final longitude is off by 2.7 millionths of a degree. The page is checked against the point he started from, 73.5° W.
The Sources
- J. P. Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395, 1987. Public domain. Retrieved from pubs.usgs.gov on 8 October 2026.
- J. P. Snyder and P. M. Voxland, An Album of Map Projections, USGS Professional Paper 1453, 1989. Public domain. Retrieved from pubs.usgs.gov on 8 October 2026.
- B. Šavrič, T. Patterson and B. Jenny, “The Equal Earth map projection”, International Journal of Geographical Information Science 33 (3), 2019: the formula and its coefficients. Test values from PROJ,
test/gie/more_builtins.gie, MIT licence, retrieved from github.com/OSGeo/PROJ on 8 October 2026. - H. Moritz, “Geodetic Reference System 1980”, Journal of Geodesy 74, 2000: the ellipsoid and its authalic radius.
- Natural Earth, 1:110m Admin 0 – Countries, version 5.1.2. Public domain. Retrieved from github.com/nvkelso/natural-earth-vector on 8 October 2026.
- d3-geo 3.1 and topojson-client 3.1, by Mike Bostock, ISC licence.
What It Misses
A sphere here, and outlines made for a whole-world view
- The Earth here is a sphere with the surface area of the GRS 80 ellipsoid. A small patch’s area on that sphere differs from its area on the ellipsoid by −0.45% at the Equator to +0.90% at the poles, and a projection on the ellipsoid, as a survey uses, has slightly different h and k.
- The outlines are Natural Earth’s 1:110m, made for a whole-world view. Every coast is simplified, so an area computed from them differs from a surveyed one; Greenland here covers 2,189,723 km². Countries too small to show at that scale are not in it: Singapore, Malta, Bahrain, Andorra, Monaco, Liechtenstein, Cape Verde and Mauritius among them. 177 countries are.
- Natural Earth draws boundaries as they are held on the ground, including disputed ones. The page takes them as they are and makes no claim about any of them.
- Robinson published a table, not a formula. Between its 5° rows the page interpolates with a cubic; another cubic would give slightly different numbers between rows.
- A country carried far turns as it goes: moving it along a great circle is a turn of the whole sphere, so its own north no longer points to the map’s north.
- Near a pole, and on the line where a map is cut, there is no single scale. The readout says so rather than print a number.