Mercator
Conformal- Area vs the equator
- 4.00×
- Scale east–west (k)
- 2.00
- Scale north–south (h)
- 2.00
- Max. angular distortion
- 0.0°
Keeps local angles, so small shapes look right, but scale grows as sec φ in every direction and area as sec² φ.
Map projection distortion
The map projection distortion calculator shows how much Mercator, Gall-Peters and Robinson distort area, scale and angles at the latitude you pick, because every flat map of the world stretches something. It is for anyone who wants to know why countries look the size they do on a map. The formulas, and the Robinson table, are on the page.
North and south give the same values. The slider stops at 85°: at the pole, Mercator's scale is infinite.
Gall-Peters
Equal-area- Area vs the equator
- 1.00×
- Scale east–west (k)
- 1.41
- Scale north–south (h)
- 0.71
- Max. angular distortion
- 38.9°
Keeps area everywhere by stretching one direction and squashing the other. Shapes are true only at 45°; at the equator they are twice as tall as wide, at 60° twice as wide as tall.
Robinson
Compromise- Area vs the equator
- 1.46×
- Scale east–west (k)
- 1.36
- Scale north–south (h)
- 0.88
- Max. angular distortion
- 24.6°
Neither conformal nor equal-area: defined by a table rather than a formula, to look right as a whole. Values here are on the central meridian.
The ellipse is Tissot's indicatrix: what a small circle on the globe becomes on the map, drawn with width k and height h (capped at 3× so it fits). The letters follow Snyder: h along the meridian, k along the parallel.
Values by latitude
Area compared with the equator of the same projection, and maximum angular distortion.
| Latitude | Mercator: area | Gall-Peters: area | Robinson: area | Gall-Peters: shape | Robinson: angles |
|---|---|---|---|---|---|
| 0° | 1.00× | 1.00× | 1.00× | 2.00 : 1 | 7.1° |
| 15° | 1.07× | 1.00× | 1.02× | 1.87 : 1 | 5.7° |
| 30° | 1.33× | 1.00× | 1.11× | 1.50 : 1 | 1.2° |
| 45° | 2.00× | 1.00× | 1.25× | 1.00 : 1 | 7.4° |
| 60° | 4.00× | 1.00× | 1.46× | 2.00 : 1 | 24.6° |
| 75° | 14.93× | 1.00× | 2.03× | 7.46 : 1 | 59.1° |
| 85° | 131.65× | 1.00× | 3.13× | 65.82 : 1 | 116.0° |
Why every flat map distorts
Gauss's Theorema Egregium (1827) shows that a surface's curvature can't change when you bend it without stretching. A sphere is curved and a sheet of paper is not, so no flat map can keep areas, angles and distances all true at once. Each projection chooses what to give up.
Mercator: true angles, growing scale
Gerardus Mercator drew his projection in 1569 for navigation: any line of constant compass bearing is a straight line on it. To keep angles true, the map must stretch north–south exactly as much as east–west, and that stretch is sec φ at latitude φ. Area grows as the square:
h = k = sec φ ⇒ area = sec² φ
At 60°, sec² φ = 4: Oslo's surroundings are drawn four times larger than the same area at the equator. At 72°, the latitude of central Greenland, the factor is about 10.5.
Gall-Peters: true area, wrong shapes
Gall-Peters is a cylindrical equal-area projection with its standard parallels at 45°. It keeps area by stretching one direction exactly as much as it squashes the other:
k = cos 45° / cos φ · h = cos φ / cos 45° ⇒ h · k = 1
At 45° a circle stays a circle. At 60° it becomes an ellipse twice as wide as it is tall (2 : 1), and at the equator one twice as tall as it is wide.
Robinson: a table, not a formula
Arthur Robinson designed his projection in 1974 by eye, to make the whole world look right rather than to keep any property exact. It is defined by a table: for every 5° of latitude, the length of the parallel and its distance from the equator. We interpolate that table with a cubic spline. Because its meridians curve, the scale also changes with longitude; the figures here are on the central meridian.
Why there is no Dymaxion column
An earlier version of this page had one. Its numbers came from a made-up formula, so we removed it. Buckminster Fuller's Dymaxion map projects the globe onto the twenty faces of an icosahedron and unfolds them. Its distortion depends on where a point falls inside a face, not on latitude: two cities on the same parallel can sit in different faces with different distortion. A single number per latitude would be wrong by construction.
What Globdrop uses
Globdrop's world maps use the Miller cylindrical projection, a softened Mercator that stretches the poles less. Its regional maps use projections suited to each area. The full list is on the Globdrop page. We wrote about what Mercator does to the size of countries in How the Mercator projection changes the size of the world, and about the maths of Globdrop's six systems in Six coordinate systems in one geography game.
Sources
- John P. Snyder, Map Projections — A Working Manual, U.S. Geological Survey Professional Paper 1395, 1987: formulas for Mercator and the cylindrical equal-area projections.
- John P. Snyder and Philip M. Voxland, An Album of Map Projections, U.S. Geological Survey Professional Paper 1453, 1989: the Robinson table. The same table is in PROJ.
Frequently asked questions
Why does every flat map distort the world?
Gauss's Theorema Egregium (1827) shows that a surface's curvature can't change when you bend it without stretching. A sphere is curved and a sheet of paper is not, so no flat map can keep areas, angles and distances all true at once. Each projection chooses what to give up.
How much does Mercator enlarge areas?
Mercator keeps angles true by stretching north–south as much as east–west, so area grows as sec² φ. At 60° latitude the factor is 4, and at 72°, the latitude of central Greenland, about 10.5.
What does Gall-Peters keep, and what does it lose?
It keeps area and distorts shape. At 45° a circle stays a circle; at 60° it becomes an ellipse twice as wide as it is tall (2 : 1), and at the equator one twice as tall as it is wide.
How are the Robinson figures worked out?
Robinson is defined by a table, not a formula: for every 5° of latitude, the length of the parallel and its distance from the equator. We interpolate it with a cubic spline. Its meridians curve, so the scale also changes with longitude; the figures on this page are on the central meridian.
Why is there no Dymaxion column?
An earlier version had one, but its numbers came from a made-up formula, so we removed it. Dymaxion's distortion depends on where a point falls inside one of the icosahedron's faces, not on latitude, so a single number per latitude would be wrong by construction.