Why can't a flat map ever get both angles and area right?

Some maps get every angle exactly right. If two roads on Earth cross at 39 degrees, a conformal map will show them crossing at exactly 39 degrees too, no matter where on the globe they are. That's why sailors and pilots rely on conformal maps: a compass bearing drawn as a straight line stays true the whole way, and on a Mercator chart, any line of constant compass direction (a rhumb line) really is a straight line you can follow.

But that perfection comes at a price, and it's not optional. In 1775, mathematician Leonhard Euler proved that a conformal map, one that preserves every angle, can never also preserve area. And an equal-area map, one that shows countries at their true relative sizes, can never be conformal either. You cannot have both on the same flat map. This isn't a limitation of current technology or cartographic skill: it follows from Carl Gauss's 1827 'Theorema Egregium,' a deep result in geometry showing that a curved surface like a sphere simply cannot be flattened without distorting either its angles or its areas somewhere.

So every conformal map quietly cheats on size. It keeps small shapes looking correctly proportioned wherever you look, but the scale, how many real kilometers one map-centimeter represents, changes from place to place. On a Mercator chart, for instance, the scale changes with latitude, which is why such maps often carry a separate scale bar for each latitude: you can't compare the sizes of two far-apart regions by eye. Mapmakers accept this trade-off deliberately. On a large-scale map of a small area, the figures are small enough to look nearly true, and charts for sailors and pilots, or weather maps of air pressure, use conformal projections because directions matter there. For maps of the whole world, where the scale varies far more, recent mapmakers tend to choose other projections instead.