If you've ever drawn a ruler-straight line between two cities on a flat world map, you've been lied to. Here's what's actually happening, why pilots fly "the wrong way," and how to think about distance on a planet that isn't flat.
The Earth is roughly a sphere. A flat map is, by definition, not. Any time you take a curved surface and squash it onto a flat rectangle, something has to give. You can preserve angles, or you can preserve areas, or you can preserve distances along certain lines — but you cannot preserve all three at once. This is a consequence of Gauss's Theorema Egregium, a theorem from 1827 that proves a sphere can't be flattened without distortion. Every flat world map you've ever seen is a compromise.
The most common compromise in everyday use — in classrooms, in newspapers, in Google Maps until you zoom out — is the Mercator projection, drawn by Flemish cartographer Gerardus Mercator in 1569. Mercator's projection preserves angles. That makes it useful for one specific job (more on that later) but the trade-off is that distances and areas get progressively more wrong as you move toward the poles.
Mercator's clever trick is that lines of constant compass bearing — what sailors call rhumb lines — appear as straight lines on his map. If you set your boat's heading to "north-northeast" and never change it, your route on a Mercator chart is a straight line. That's a genuinely useful property for marine navigation in the age of magnetic compasses and no GPS.
The price is that everything north of the tropics gets stretched. The math literally divides by the cosine of the latitude — at 60°N (the rough latitude of Oslo or Anchorage) features are stretched by a factor of two. At 80° the stretch is nearly six. This is why on a Mercator map:
The shortest path between two points on a sphere lies along a great circle: any circle on the surface of the sphere whose center coincides with the center of the sphere. The equator is a great circle. So is every line of longitude. Most great circles, though, are tilted relative to those — they slice the planet at some angle that doesn't align with our familiar grid.
If you take a globe (a real, physical one) and stretch a piece of string tight between two cities, the string traces a great circle. That's the route a plane (or, in principle, a missile, or a fiber-optic cable) wants to follow. It's the path that minimises distance on a sphere.
On a Mercator map, that same path almost always looks curved, often curving toward the nearest pole. People assume the curve means the route is going "out of its way." It isn't. The map is going out of its way. The globe is the truth.
On a flat Mercator map a line from Oslo (≈60°N) to Tokyo (≈36°N) looks like it should go more or less due east, sweeping across Europe, the Caspian region, and the steppes of Central Asia. The real flight path doesn't. It heads northeast, climbs over the Arctic, and comes down through Siberia. The great-circle distance is about 8,400 km. The Mercator-ruler "distance" between the same two points reads about 11,800 km — a 40% overshoot.
Pilots have been flying this kind of route since long-haul jet aviation became routine. It just looks weird on the maps we grew up with.
Looks like a straight Atlantic crossing on a Mercator map. The real shortest path arcs noticeably north — passing close to Newfoundland and the Azores rather than going straight east.
Looks like an unbroken Pacific crossing on Mercator. The real great-circle path bends way south, dipping into the Southern Ocean toward Antarctica. That's why some long-haul southern-hemisphere flights pass surprisingly far south — and why a Mercator map drawn from a single hemisphere makes them look like they're taking a strange detour.
| Route | True (great-circle) | Mercator straight-line | Error |
|---|---|---|---|
| Oslo → Tokyo | ~8,400 km | ~11,800 km | +40% |
| Anchorage → Helsinki | ~6,800 km | ~9,400 km | +38% |
| Los Angeles → London | ~8,800 km | ~11,000 km | +25% |
| Cape Town → Sydney | ~11,000 km | ~12,300 km | +12% |
| Quito → Singapore (near equator) | ~19,400 km | ~19,500 km | ~0% |
Notice the pattern: the further from the equator the cities lie, the larger the Mercator error. Pairs that sit on or near the equator agree closely, because Mercator's distortion is essentially zero at the equator and grows toward the poles.
So if Mercator gets distance wrong, why did sailors use it for centuries? Because in pre-GPS marine navigation, the practical question wasn't "what's the shortest path?" — it was "what's the path I can actually steer?" A great-circle route requires you to constantly change compass heading, sometimes by tens of degrees. That's hard to do on a wooden ship with a magnetic compass and a sextant. A rhumb line — the straight line on Mercator — has a single constant heading. You set your sails, tie the wheel, and the ship walks the line. Slightly longer in distance, vastly easier in practice.
Modern aviation has the opposite problem. Aircraft can adjust heading continuously, fuel matters more than steering simplicity, and the saved kilometres translate directly into saved fuel and time. So flights fly great circles, even though they look like detours.
Distance is just one of Mercator's distortions. Area is the other big one, and it has cultural consequences worth knowing about. Because countries near the poles are inflated, the relative size of nations on standard world maps is misleading in ways that subtly shape how people picture the world.
The "True Size" tool — and many similar online toys — let you drag countries around to see their real proportional sizes on the same Mercator map. It's an eye-opening exercise.
If Mercator is so problematic, why is it still everywhere? Mostly inertia, plus its mathematical convenience for tile-based web maps (Google Maps' tiles are square at every zoom level because of Mercator). But cartographers have produced dozens of alternatives, each with its own trade-offs.
For the curious, the great-circle distance between two latitude/longitude points (φ₁, λ₁) and (φ₂, λ₂) on a sphere of radius R is given by the haversine formula:
a = sin²(Δφ/2) + cos(φ₁) · cos(φ₂) · sin²(Δλ/2) c = 2 · atan2(√a, √(1−a)) d = R · c
For Earth, R ≈ 6,371 km. Plug in the latitudes and longitudes in radians and you get distance in kilometres. This formula handles even antipodal points well — better than the simpler "law of cosines" form which suffers from floating-point precision loss for nearby points.
Mercator's projection, by contrast, maps latitude φ to a vertical coordinate y via:
y = ln(tan(π/4 + φ/2))
That logarithm is what makes near-polar latitudes blow up — and why every Mercator map you've ever seen cuts off the poles at around ±85°.
The Mercator projection is a marvellous mathematical achievement and an excellent navigation chart. It's also a poor general-purpose map of the world, because the trade-offs it makes — preserving angle at the cost of area and distance — happen to be the trade-offs least aligned with the things most people want from a world map: a sense of how big places are and how far apart they sit.
The good news is, the modern web makes the alternative easy. You can spin a real 3D globe in any browser, drop two cities on it, and see the actual great-circle path bend across the planet — no flat-map distortion, no mental gymnastics. That's what this site is for.
Try it on the globe →