Great Circle vs Rhumb Line: The Shortest Path on Earth
On a sphere, the shortest path is a great circle. A rhumb line holds a constant bearing but is longer. Learn the difference.
When plotting a course between two points on Earth, you have two classical options: the great-circle route, which is the shortest possible path, or the rhumb line, which holds a constant compass bearing but runs longer. Pilots, sailors and map-makers have argued about the trade-off for centuries. This guide explains both, shows how different they really are with numbers, and when each is the right choice.
Great circles: the shortest path
A great circle is any circle on the sphere whose centre coincides with the Earth's centre β the equator and all meridians are great circles. Any two points that are not exactly opposite each other (antipodal) define exactly one great circle, and the shorter arc between them is the shortest path along the surface. This is the route airlines fly and the distance your 'as the crow flies' calculator reports, computed with the Haversine formula. The counter-intuitive part: on the flat Mercator maps most of us grew up with, great-circle routes look curved β arcing toward the poles β because the map itself is distorted, not because the route is.
Rhumb lines: the constant-bearing path
A rhumb line (or loxodrome) crosses every meridian at the same angle β meaning you can sail or fly it by holding one constant compass heading the whole way. On a Mercator projection, a rhumb line is a perfectly straight line, which is exactly why Mercator designed his projection for navigators in 1569: plot a straight line, read off the bearing, and steer that course. The price is distance: except along the equator or a meridian, the rhumb line is longer than the great circle, sometimes dramatically so. Before computers, the simplicity of 'one heading all the way' was worth the extra miles.
How different are they? Worked numbers
For short hops the difference is negligible. Take London to Paris (about 340 km): the great circle and the rhumb line differ by well under a kilometre β irrelevant. But stretch the route and the gap grows. London to Tokyo is about 9,560 km by great circle; the constant-bearing rhumb line is roughly 11,300 km β about 18% longer, or nearly 1,800 extra kilometres of fuel. The extreme case is an east-west route at high latitude: along 60Β° N, the rhumb line between two points 90Β° of longitude apart is about 1.57 times the great-circle distance. Rule of thumb: the longer the route and the more it runs east-west at high latitude, the more the great circle wins.
When the two agree
The routes coincide in two cases: travelling due north-south along a meridian (both are the meridian itself), and travelling due east-west along the equator (both are the equator). Near these cases they are close: short routes, north-south routes, and routes near the equator show little difference. This is why the distinction rarely matters for local navigation β driving directions, day hikes, city wayfinding β and matters enormously for intercontinental aviation and ocean shipping, where the routes are long and often run east-west across latitudes.
What aviation and shipping actually do
Modern aviation flies great-circle routes β but with a practical twist. A true great circle requires constantly changing heading, which is tedious for pilots and autopilots alike. So long routes are broken into a series of short rhumb-line segments between waypoints: each leg holds a constant heading, and the legs together approximate the great circle. Air traffic control, winds and restricted airspace modify the ideal path further. Ocean shipping does the same, with an added consideration: the shortest path sometimes crosses dangerous weather or ice, so routers optimise for time, fuel and safety rather than pure distance β the great circle is the starting point, not the final answer.
A short history
For most of the age of sail, navigators used rhumb lines because they were computable with a compass and a Mercator chart β great-circle sailing required spherical trigonometry that was impractical at sea. The mathematics was known (it is just spherical geometry), and 19th-century navigators developed 'composite sailing' and great-circle charts for the longest ocean passages, notably on the clipper routes. Aviation made great circles mainstream: once aircraft could fly thousands of kilometres non-stop, the fuel savings were too large to ignore, and flight computers made the math trivial. Today every flight-planning system and every distance calculator uses great circles by default.
Computing them yourself
The great-circle distance comes from the Haversine formula: with latitudes Ο1, Ο2 and longitude difference ΞΞ», compute a = sinΒ²(ΞΟ/2) + cos Ο1 Β· cos Ο2 Β· sinΒ²(ΞΞ»/2), then distance = 2RΒ·arcsin(βa) with R = 6,371 km. The initial great-circle bearing comes from spherical trigonometry (atan2 of a standard expression). The rhumb-line distance is the meridian-arc difference divided by the cosine of the mean latitude, adjusted properly near the poles. You do not need to implement these by hand β a distance calculator does the great-circle math, and a bearing calculator gives the initial heading β but knowing which one you are looking at explains why the 'straight line' on your map app sometimes looks curved.
Polar routes: where the arc really matters
The most dramatic great-circle savings appear on polar routes. A flight from New York to Hong Kong via the great circle arcs up over the Arctic β far north of the 'straight line' most people imagine on a flat map β cutting hundreds of kilometres off the rhumb-line distance. These polar routings became routine only when aircraft gained the range and cold-weather reliability to use them, and when ETOPS regulations (which govern how far twin-engine aircraft may fly from diversion airports) evolved to permit them. For passengers, the visible sign is the in-flight map: the aircraft icon appears to swing far north before curving back down. That arc is not a detour; it is the shortest path, and the map is simply telling the truth about a curved Earth on a flat screen.
Try the math
Run the numbers yourself with the great-circle distance calculator, read our guide to measuring distance between cities, or estimate flight time for a route you care about.
Frequently asked questions
Is a flight path a straight line on a flat map?
No β it looks curved on most flat maps because it follows a great circle on the globe, and the map projection distorts it. On a globe, the same route is the 'straightest' possible line.
Why did old sailing charts prefer rhumb lines?
A navigator could steer a single constant compass bearing for the whole voyage without doing spherical trigonometry. The Mercator projection was designed specifically to make rhumb lines straight. Computers removed that constraint.
Do ships and planes follow great circles exactly?
Approximately. Real routes are great circles broken into waypoint legs, then adjusted for winds, currents, weather, air traffic and restricted zones. The great circle is the baseline that gets optimised, not the final track.
Why do flight maps show planes flying over Greenland to get to Asia?
Because the great-circle route from eastern North America to East Asia passes near or over the Arctic. On a flat map it looks like a detour north; on the globe it is the shortest path.
Do I ever need rhumb lines today?
For everyday navigation, no β GPS and routing apps handle great-circle math invisibly. Rhumb lines survive mainly as a teaching concept, in traditional celestial navigation, and as the short legs that approximate great-circle routes between waypoints.
Sources & data
Authoritative references used to research and verify this article: