It is hard to represent our spherical world on a flat piece of paper. Cartographers use something called a "projection" to morph the globe into a 2D map. The most popular of these is the Mercator projection.
The Peters projection shows approximately the same areas, but it does so by distorting latitudes. In doing so, it distorts the proportions of nations and continents, so the size is closer to reality, but the shape is wrong. The whole concept of mapping a sphere on flat space is too intrinsically messed up, to begin with.
Every map projection introduces distortion, and each has its own set of problems. One of the most common criticisms of the Mercator map is that it exaggerates the size of countries nearer the poles (US, Russia, Europe) while downplaying the size of those near the equator (the African Continent). On the Mercator projection, Greenland appears to be roughly the same size as Africa. In reality, Greenland is 0.8 million sq. miles and Africa is 11.6 million sq. miles, nearly 14 and a half times larger.
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