Great Circle vs Rhumb Line: Which Route Is Shorter?
A great circle is the shortest path between two points on a sphere. A rhumb line is the easiest to navigate. They are not the same line. On a short hop the difference is academic. On a transatlantic the difference is sixty nautical miles. On a transpacific the difference is several hundred. Here is the math, the history, and why every modern flight management computer flies great circles whether the pilot is thinking about it or not.
The Two Definitions
A great circle is any circle on the surface of a sphere whose center coincides with the center of the sphere. The shortest path between any two points on a sphere is the arc of the unique great circle that connects them. On Earth, that arc is the shortest possible track from A to B.
A rhumb line, also called a loxodrome, is a path of constant compass bearing. Pick any heading, hold it, and you are flying a rhumb line. The line crosses every meridian at the same angle. It is the path of least mental effort. It is also longer than the great circle for any pair of points not on the same meridian or the equator.
On a globe, the great circle looks like an arc bending toward the pole. The rhumb line looks like a smooth spiral that eventually corkscrews into the pole if extended forever. On a Mercator projection, the relationship inverts. The rhumb line becomes a straight line. The great circle becomes a curve.
Why Mercator was invented. Gerardus Mercator published his projection in 1569 specifically so that sailors could navigate by drawing a straight line on a chart, reading the constant bearing, and holding that course. For a sextant-and-compass crew on a slow ship, the rhumb line was the operationally correct route even though it was longer. Mercator's projection turned navigation into a ruler-and-protractor problem. Five hundred years later, that same convenience is why people still find great-circle routing counterintuitive when they look at a flat map.
Why They Look So Different on a Flat Map
Mercator preserves angles. It does not preserve area or distance. Areas near the poles get stretched horizontally. Greenland looks the size of Africa. Antarctica looks the size of every other continent combined.
The same stretching distorts how routes appear. A great circle from New York to Tokyo passes over the Bering Strait. On a globe, that is obviously the shortest path. On Mercator, it appears to swing wildly north before curving back down, which looks wasteful and is the opposite of true. The rhumb line, plotted as a straight line southwest-to-northeast across the middle of the Pacific, looks shorter on the chart but actually is longer in reality.
The bias is geometric, not visual. Any flat projection of a sphere must distort either angle, area, distance, or shape. Mercator chose to preserve angle for navigation. Equal-area projections preserve area for thematic mapping. Orthographic projections preserve the appearance of a globe from a single viewpoint. None of them can show a great circle as a straight line and a rhumb line as a straight line at the same time.
The Math, in Brief
Great-circle distance uses the Haversine formula. Given two points in latitude and longitude on a sphere of radius R, the great-circle distance is:
d = 2R · arcsin(√(sin²(Δφ/2) + cos φ₁ · cos φ₂ · sin²(Δλ/2)))
Great Circle Pro uses an Earth radius of 3,440.065 nautical miles. The initial bearing along the great circle is computed from the forward azimuth between origin and destination. That bearing changes continuously as you travel along the arc, which is why you cannot just "fly the heading and hope" on a great circle. The flight management computer recomputes the desired track every few seconds.
Rhumb-line distance uses the loxodrome formula, derived from Mercator's projection. The bearing is constant for the entire route. The distance is always longer than the corresponding great circle, with the gap growing as the route lengthens and as it tilts away from the equator and away from a single meridian.
When the Difference Matters
Real numbers from common city pairs. Distances rounded to whole nautical miles. Great-circle figures are exact; rhumb-line figures are computed from the loxodrome formula on a spherical Earth model.
| Route | Great Circle | Rhumb Line | Delta |
|---|---|---|---|
| ATL → MIA | 515 nm | 515 nm | 0 nm |
| JFK → LAX | 2,144 nm | 2,182 nm | 38 nm |
| JFK → LHR | 3,005 nm | 3,109 nm | 104 nm |
| LAX → SYD | 6,510 nm | 6,668 nm | 158 nm |
| JFK → HKG | 7,019 nm | 7,742 nm | 723 nm |
| NRT → JNB | 7,512 nm | 8,491 nm | 979 nm |
A short hop at low latitudes that runs east-west, ATL to MIA, the difference is essentially zero. A coast-to-coast US route picks up roughly forty nautical miles. A transatlantic gives back about a hundred. A polar or near-polar transpacific can save seven hundred nautical miles or more.
Seven hundred nautical miles at five hundred knots of groundspeed is eighty-four minutes of flight time. It is several tonnes of fuel. It is the difference between a route that closes and a route that needs an intermediate fuel stop. The geometry compounds with the economics in a way that gets serious very quickly.
Why Pilots Stopped Hand-Flying Rhumb Lines
Modern flight management systems compute great-circle tracks continuously. The autopilot follows the FMS-commanded track, which is the great circle for the active leg. The bearing the pilot sees on the heading bug is what the great-circle track happens to be at that instant, and it drifts slowly as the aircraft moves along the arc.
A pre-FMS aircraft could not fly a great circle directly. The crew would break a long over-water route into a series of straight rhumb segments, each one a few hundred miles long, and step-change the heading at each waypoint. The composite path approximated the great circle without ever actually flying it. The fuel cost of those approximations was significant but tolerable in an era when the marginal cost of fuel was lower and the alternative was unreliable celestial navigation.
GPS and ring-laser inertial navigation made true great-circle tracking trivial. The aircraft knows where it is to within a few meters and recomputes its desired track ten times a second. The fuel savings from flying the actual great circle, especially on transpacific and transpolar routes, pay back the avionics in roughly the time it takes to make a sandwich.
The Classic Examples
New York to Hong Kong
On a Mercator chart the obvious-looking route is a straight southwesterly line across the middle of the Pacific. The true great-circle route bends north and arcs over the Beaufort Sea, the East Siberian Sea, and Sakhalin before descending into Hong Kong. The great-circle saves over seven hundred nautical miles and crosses some of the most remote airspace on Earth.
This route is also a poster child for the ETOPS framework. The polar arc passes through regions with minimal alternate density, which is exactly the problem ETOPS-330 and ETOPS-370 ratings were designed to solve.
Anchorage to Frankfurt
Anchorage to Frankfurt is roughly 4,400 nm great-circle, with the arc passing over the high Arctic just north of Greenland. The corresponding rhumb line bends substantially south through eastern Canada and across the North Atlantic, adding about two hundred nautical miles. The great-circle is also closer to the regions with adequate diversion airports for high-latitude operations, which is why Anchorage was historically a refueling stop for trans-polar cargo before the 747 made it nonstop in either direction.
Sydney to Buenos Aires
A trans-South-Pacific route in the Southern Hemisphere with no major land between origin and destination. The great-circle arcs deep into the southern ocean, crossing within a thousand miles of the Antarctic ice edge. The rhumb-line track stays roughly along 40°S, which looks similar at first glance but adds roughly three hundred nautical miles. The route is also one of the few commercial pairings where overflight options are essentially nil. The geometry, the wind, and the ETOPS exposure all conspire to make it a non-trivial planning exercise.
How Great Circle Pro Shows Both
Every route on the map renders as a great-circle arc. The Flight Plan card shows the great-circle distance and, for reference, the rhumb-line distance and the delta between them. On short routes the delta is small. On long routes the delta is a planning signal.
- Build a route in the main map tool by typing origin and destination ICAO or IATA codes.
- Switch the projection between Mercator and Orthographic to see how dramatically the visual shape of the route changes while the actual distance does not. The Orthographic globe view is where the great-circle line finally looks like the straight line it actually is.
- Compare against the rhumb line by reading the delta on the route card. Anything over fifty nautical miles is worth a second look.
- Use the Azimuthal Equidistant projection centered on your origin airport. Every distance on that projection is true scale from the center, so the great-circle arc draws as a straight radial line. It is the cartographically honest view of "what is reachable from here."
Polar routes for the curious. Try JFK to HKG, ATL to BLR, or DFW to DXB on the Orthographic projection. The route lines arc over the Arctic in a way that looks impossible on a Mercator map and obvious on a globe. That is the conceptual unlock. Once you internalize it, every long-haul route on a flat map starts to look slightly wrong.
The One-Sentence Summary
A great circle is the shortest distance. A rhumb line is the easiest bearing. Modern aircraft fly great circles because GPS and FMS made the easy bearing irrelevant, and the fuel savings on long-haul routes are enormous. Flat maps will keep lying to your intuition about which is shorter, and globes will keep showing you the truth.