Great Circle Distance (Haversine Formula)
The great circle distance is the shortest path between two points on a sphere — the route followed by aircraft and long-distance sailing. It differs from road distance.
Haversine Formula
a = sin²(Δlat/2) + cos(lat₁)·cos(lat₂)·sin²(Δlon/2)
c = 2·atan2(√a, √(1-a))
d = R·c (R = 6371 km, Earth's mean radius)
Δlat = lat₂ - lat₁ (in radians)
Δlon = lon₂ - lon₁ (in radians)Example: London to New York
London: 51.51°N, 0.13°W
New York: 40.71°N, 74.01°W
Great circle distance ≈ 5,570 km (3,460 miles)
By air: ~7 hours eastbound, ~6.5 hours westbound
(Jet stream adds ~30 min westbound)Distance vs Travel Time Estimates
- Walking: 5 km/h → 1km ≈ 12 minutes
- Cycling: 20 km/h → 10km ≈ 30 minutes
- Driving (city): 30-40 km/h average
- Driving (motorway): 100-130 km/h
- Commercial flight: 800-900 km/h (ground speed)
Calculate geographic distance: Free Distance Calculator
Geographic Distance Quick-Reference Table
| Route | Great-circle distance | Typical flight time |
|---|---|---|
| New York → Los Angeles | 3,940 km (2,449 mi) | ~5.5 hours |
| London → New York | 5,571 km (3,461 mi) | ~7 hours |
| Sydney → London | 16,993 km (10,562 mi) | ~21 hours |
| Dubai → Singapore | 5,843 km (3,630 mi) | ~7 hours |
| Paris → Tokyo | 9,714 km (6,036 mi) | ~12 hours |
| Cape Town → Mumbai | 6,254 km (3,886 mi) | ~8 hours |
How Geographic Distance Calculation Works
The Haversine formula calculates the great-circle distance (shortest path on a sphere) between two points given their latitude and longitude: d = 2R × arcsin(√[sin²(Δlat/2) + cos(lat₁)cos(lat₂)sin²(Δlon/2)]), where R = 6,371 km (Earth's mean radius). This is the straight-line "as the crow flies" distance. Actual travel distances by road or rail are always longer due to terrain, road networks, and infrastructure.
Latitude runs north-south (−90° to +90°); longitude runs east-west (−180° to +180°). Each degree of latitude = ~111 km (invariant). A degree of longitude varies: ~111 km at the equator, shrinking to 0 at the poles (multiply by cos(latitude)). GPS coordinates are expressed as decimal degrees (e.g., 51.5074, −0.1278 for London) or degrees-minutes-seconds (DMS). The WGS84 ellipsoid (used by GPS) models Earth as an ellipsoid for higher precision than a perfect sphere.
Common Mistakes
- Confusing great-circle distance with driving distance: The straight-line distance from London to Edinburgh is ~534 km; the driving distance via motorway is ~640 km. For cross-country or inter-city travel planning, always use road/route distance calculators, not great-circle distance.
- Swapping latitude and longitude: Latitude is the north-south coordinate (first, in most APIs); longitude is east-west (second). Reversing them places your point on the wrong side of the planet — a common error in mapping APIs and spreadsheet geocoding.
- Using the wrong Earth radius: Earth's radius at the equator is 6,378 km; at the poles, 6,357 km; mean is 6,371 km. For precision applications (surveying, navigation), use the WGS84 ellipsoid parameters rather than a sphere assumption.
Frequently Asked Questions
Flat maps (Mercator projection) distort great-circle routes — they appear curved on the map but are actually the shortest path on the globe. A flight from London to Tokyo goes north over Russia (not east directly) because the great-circle route across the northern latitudes is shorter than following the same latitude eastward. Flight paths also account for jet streams (strong high-altitude winds), restricted airspace, and emergency diversion airports.
Euclidean distance is the straight-line distance in 3D space (through the Earth). Geodesic distance is the shortest surface path on a sphere or ellipsoid — what a plane actually flies. For small distances (city-level), both are nearly identical. For intercontinental distances, Euclidean (chord) distance is slightly shorter than geodesic because the chord cuts through the Earth while the geodesic follows the curved surface.
Standard civilian GPS accuracy is 3–5 metres under open sky. Modern smartphones with GPS + GLONASS + Galileo achieve 2–3 m accuracy. In urban canyons (tall buildings blocking satellite signals), accuracy degrades to 10–30 m. Differential GPS (DGPS) and RTK (Real-Time Kinematic) systems used in surveying achieve centimetre accuracy. Distance measurements accumulate these errors — a GPS-tracked run of 10 km may be accurate to ±50 m under good conditions.