How this instrument works
A great circle is the largest circle you can draw on a sphere: the line where the sphere's surface meets a flat plane that passes exactly through its center. Every meridian of longitude is a great circle, and so is the equator, but no other line of latitude qualifies — a circle traced around 51°N sits on a plane that misses the center by a wide margin, so its own distance-around is smaller than the sphere's true circumference. Constrained to move along the surface, the shortest route between any two points on a sphere always follows the arc of the great circle that joins them, never a straight chord through the interior.
The arc length itself is old geometry: distance equals radius times the angle the two points make at the center, measured in radians. Everything this calculator does is aimed at recovering that one angle accurately. A direct spherical law of cosines can find it, but taking an arccosine of a value pressed close to 1 — which happens whenever the two points sit near each other — amplifies tiny rounding errors badly. The haversine form sidesteps this by working with sin²(Δφ⁄2) and sin²(Δλ⁄2) instead, quantities that stay well clear of that unstable region, so the angle comes out reliable whether the points are a mile apart or half a world apart.
The model treats Earth as a perfect sphere of radius 6,371,000 metres, its mean radius once the equatorial bulge and polar flattening are averaged out. That approximation is accurate to roughly 0.3% against the true oblate shape — plenty for navigation, logistics, or curiosity, though land surveyors reach for an ellipsoidal method such as Vincenty's formula when centimetres matter. At the extremes, two identical coordinates return a distance of exactly zero, and two points on opposite sides of the globe return the largest value the formula can ever produce: radius times π, about 20,015 km.
- Enter the first location's coordinates in Latitude 1 and Longitude 1, in decimal degrees with north and east positive.
- Enter the second location's coordinates the same way, in Latitude 2 and Longitude 2.
- Read Great-circle distance for the shortest path over Earth's curved surface between the two points.
- Switch the distance unit between kilometres, miles, and metres to match how you need to report the figure.
- Order does not matter — swapping point 1 and point 2 returns the identical distance.
Worked example — New York to London
New York sits at 40.7128°N, 74.0060°W and London sits at 51.5074°N, 0.1278°W. Entered as Latitude 1, Longitude 1, Latitude 2, and Longitude 2, the sheet first converts each figure to radians, then works out Δφ and Δλ between the two cities before folding them into the haversine formula.
The result is a Great-circle distance of 5,570,222.18 metres — 5,570.22 kilometres — the shortest path a rope stretched over the planet's curved surface would take between the two cities. That is noticeably shorter than a straight tunnel bored through the Earth would be, and it is exactly why the flight track drawn on a flat map bows northward over the Atlantic instead of running as a straight line: the curve is the shortest route, not a detour.
Questions
What exactly is a great circle?
It is the largest circle obtainable on a sphere: the intersection of the sphere with any flat plane that passes through its exact center. Every meridian of longitude is one, and so is the equator, but no other parallel of latitude is — those smaller circles sit on planes that miss the center, so a lap around 51°N is shorter than the sphere's true circumference.
Why use half-angle sines instead of a simpler cosine rule?
The plain spherical law of cosines, cos(d⁄R) = sinφ₁sinφ₂ + cosφ₁cosφ₂cos(Δλ), takes an arccosine of a number pressed close to 1 whenever the two points are near each other, and small rounding errors blow up badly right there. The haversine form keeps every intermediate value away from that unstable zone, so nearby points still resolve accurately.
Why does the calculator use 6,371,000 metres for Earth's radius?
That is Earth's mean radius, averaged over its true shape, which bulges slightly at the equator and flattens at the poles rather than forming a perfect sphere. Treating the planet as a sphere of this radius keeps the arithmetic simple and stays accurate to roughly 0.3%; survey-grade work instead uses an ellipsoidal method such as Vincenty's formula.
Is great-circle distance the same thing as flight distance?
It is the theoretical floor — the shortest possible path over the surface, ignoring wind, air corridors, and airspace restrictions. Real long-haul routes track the great circle closely, which is exactly why they curve on a flat map: a straight line drawn on a Mercator projection is not the shortest surface path, however straight it looks.
What is the farthest apart two points on Earth can be?
Antipodal points, sitting on exactly opposite sides of the globe, are farthest apart, at a distance of radius times π — about 20,015 km with this calculator's Earth radius. That is also the ceiling the formula can ever return; nothing on a sphere's surface lies farther from a given point than its antipode.
Does swapping the two points change the answer?
No. Swapping point 1 and point 2 only flips the sign of Δφ and Δλ, and the formula squares both differences immediately, so the sign cancels out. New York to London returns the exact same figure as London to New York.