Great-circle distance, bearing, midpoint, and flight time between two coordinates.
Spherical-Earth approximation (mean radius 6371 km). Accurate to ~0.5 % vs the WGS-84 ellipsoid for typical inter-city distances. Bearing is the initial true course; on a great circle the bearing changes continuously along the route.
Computing the great-circle distance between two points on Earth from their latitude and longitude is the foundational geographic-distance problem. The naive Pythagorean approach (square root of latitude-squared + longitude-squared) is wrong because Earth is a sphere — a degree of longitude near the poles is far smaller than near the equator. The haversine formula gives the exact great-circle distance on a sphere, accurate to within 0.5 % across all latitudes (the small error comes from Earth being a slightly oblate spheroid, not a perfect sphere). This calculator computes great-circle distance between two cities, plus the initial bearing (compass direction at start), the midpoint coordinates, and an estimated commercial-aircraft flight time at 900 km/h cruise.
The haversine formula:
a = sin²(Δφ / 2) + cos φ₁ · cos φ₂ · sin²(Δλ / 2)
c = 2 · atan2(√a, √(1 − a))
d = R · c
where φ is latitude, λ is longitude (both in radians), Δφ = φ₂ − φ₁, Δλ = λ₂ − λ₁, and R is Earth's mean radius (6 371 km).
Initial bearing:
y = sin Δλ · cos φ₂
x = cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ
θ = atan2(y, x), normalized to [0, 360)
Midpoint (along the great circle):
Bx = cos φ₂ · cos Δλ
By = cos φ₂ · sin Δλ
φm = atan2(sin φ₁ + sin φ₂, √((cos φ₁ + Bx)² + By²))
λm = λ₁ + atan2(By, cos φ₁ + Bx)
Flight time at 900 km/h: distance / 900 hours; rendered as h:m:s.
The chart shows the two points and the great-circle line on a Mercator-style world map (or a log-scale reference bar comparing the distance to canonical reference distances: city block, marathon, Paris-Nice, Atlantic crossing).
Enter the start latitude and longitude (decimal degrees, positive north / east, negative south / west). Enter the end latitude and longitude. The result panel shows the great-circle distance in km and miles, the initial bearing in degrees, the midpoint coordinates, and the estimated flight time at cruise.
Paris (48.8566 N, 2.3522 E) to New York (40.7128 N, −74.0060 W):
Tokyo (35.6762 N, 139.6503 E) to Sydney (−33.8688 S, 151.2093 E):
Same hemisphere, short distance: Paris (48.8566 N, 2.3522 E) to London (51.5074 N, −0.1278 W):
Earth isn't a sphere — it's an oblate spheroid. Equatorial radius 6 378 km, polar 6 357 km. The haversine using mean radius 6 371 km is good to ~ 0.5 %. For better accuracy, use Vincenty's formulae or the WGS-84 ellipsoid model.
Sign convention. Decimal degrees, positive N/E, negative S/W. Mistakes here flip results to the antipode.
Antipodal-point edge case. When two points are nearly antipodal, the haversine has numerical instability. The calc uses atan2 which is robust, but extreme cases lose precision.
Great circle vs rhumb line. Great-circle is the shortest spherical path; rhumb-line is the constant-bearing path used in navigation. They differ — great-circle bends toward the pole. Aviation uses great-circle; small-craft sailing often uses rhumb-line for simplicity.
Initial vs final bearing. The bearing changes along the great circle. The calc gives the initial bearing at the start point. The bearing at the destination ("final bearing") is generally different.
Midpoint definition. The calc computes the great-circle midpoint. There are alternative midpoints (rhumb-line midpoint, geodesic-arc midpoint on ellipsoid).
Flight-time estimate. 900 km/h is a typical commercial-jet cruise. Real flights add ~ 30 minutes for taxi + takeoff + landing + approach delays. The number is straight-line distance / cruise speed — a lower bound.
Curvature of magnetic vs geographic north. Bearings are geographic (true). For magnetic-compass navigation, subtract the local magnetic declination (typically ± 10° in mid-latitudes).
Curvature into 3D. The calc gives 2D great-circle distance on the surface. For aircraft cruising at ~ 11 km altitude, the actual 3D distance is barely different (< 0.2 %) but for satellite computations it matters.
Datum mismatch. Lat/lon assumes WGS-84 datum (the modern standard). Older NAD-27 or local datums shift coordinates by tens of meters; usually fine for casual use, problematic for surveying.
Anti-meridian crossing. A path from 170° E to −175° W (= 185° E) is 15° westward, not 345° eastward. The calc handles this correctly because it uses cos/sin/atan2 which wrap naturally; verify with a known case.