Why all maps of the world lie

Find out why a 3D sphere cannot be perfectly projected onto a flat map. An explanation of Gauss's theorem and Mercator's distortions.

3 min readSources: 5

Flattening a sphere: the mapmaker's dilemma

Try to lay an orange peel flat. It cracks, tears or wrinkles. Mapmakers face the same problem: the Earth is roughly a sphere, while a map is a flat sheet of paper or a screen. A sphere cannot be transferred to a plane without any distortion at all.

Carl Friedrich Gauss explained why. In his 1828 work on curved surfaces he showed that a surface's curvature does not change when the surface is only bent, without stretching. This result is known as the Theorema Egregium, the “remarkable theorem”. A sphere is curved and a plane is not, so no world map can keep all distances. It can keep something else instead, such as angles in small areas or the proportions of areas. Every flat map is therefore a compromise, and the mapmaker chooses what matters most for a given use.

Quick check

Which is larger in reality: Africa or Greenland?

Mercator: a map for sailors, not for comparing countries

The best-known projection, the Mercator projection, was created in 1569 for navigation. Gerardus Mercator built the grid so that lines of constant bearing, called rhumb lines, are straight. A sailor could lay down a ruler, read the angle and hold it for the whole voyage.

The price is stretching that grows with distance from the Equator. Scale increases with the secant of latitude. At 60°, where Oslo and Helsinki lie, distances are doubled and areas are four times larger. At 80°, distances are almost six times larger and areas more than 30 times larger. The USGS album of map projections notes that Mercator is often and inappropriately used as a world map in atlases and on wall charts, because it gives a misleading view of area.

The biggest illusions

Greenland and Africa: on a Mercator map they look similar. In reality Greenland covers 2.17 million km² and Africa about 30.4 million km², roughly 14 times more. Greenland is also smaller than India (3.29 million km²). Check it in the Poland and Greenland comparison.
Russia: Russia is the largest country in the world (17.1 million km²), yet Africa is almost twice its size. On a Mercator map Russia looks much bigger because it lies far to the north.
Countries near the Equator: Brazil (8.51 million km²) and the Democratic Republic of the Congo (2.35 million km²) are very large, but near the Equator Mercator hardly enlarges them. Next to the stretched northern countries they look smaller than they really are.

Why does it matter?

John P. Snyder, author of the map projection manual published by the USGS, warned that when Mercator is the main basis of the world maps students see, it leads to mistaken ideas about the size of areas. For decades there has also been a debate about whether such maps diminish the countries of Africa and South America. In 2026 the UN General Assembly adopted a resolution encouraging equal-area maps wherever the sizes of continents are compared. We describe it in The UN's new world map.

Want to test your own intuition? Take the size illusion quiz and see how much maps have shaped your sense of scale.