Quick Reference: Converting Latitude/Longitude Differences to Meters
Converting angular coordinate differences into metric distances is a routine task in navigation, mapping, and geospatial engineering. This post serves as a self-contained reference, covering three cases in increasing generality: meridional distance (latitude-only change), parallel distance (longitude-only change), and the general two-point case via the Haversine formula.
Earth Model & Notation
All derivations here treat the Earth as a sphere of mean radius
\[R = 6{,}371{,}000 \text{ m}\]This introduces errors on the order of 0.3 % relative to a WGS-84 ellipsoid — acceptable for most engineering work at scales below a few hundred kilometres. Where sub-metre accuracy is required, consult ellipsoidal formulae (Vincenty, Karney).
| Symbol | Meaning |
|---|---|
| \(\varphi\) | Geodetic latitude (radians, positive North) |
| \(\lambda\) | Longitude (radians, positive East) |
| \(\Delta\varphi,\,\Delta\lambda\) | Small differences in latitude / longitude |
| \(R\) | Mean Earth radius (6 371 000 m) |
1 — Distance Along a Meridian (Latitude Difference)
\[\boxed{d_{\varphi} = R \,|\Delta\varphi|}\]Assumption. The two points share the same longitude, or their longitude separation is negligible compared with their latitude separation. Under the spherical model, a meridian is a great circle of radius \(R\), so arc length is simply proportional to the subtended angle.
where \(\Delta\varphi\) is in radians. In terms of decimal degrees \(\Delta\varphi_{\circ}\):
\[d_{\varphi} = R \cdot \frac{\pi}{180}\,|\Delta\varphi_{\circ}|\]Scale factor. One degree of latitude ≈ 111 139 m (111.1 km), and this value is constant across all latitudes under the spherical assumption — meridians are all great circles of the same radius.
Latitude difference → metres
Uses d = 6 371 000 × |Δφ| × π/180
2 — Distance Along a Parallel (Longitude Difference)
Assumption. The two points share the same latitude \(\varphi\), or their latitude separation is negligible. A parallel at latitude \(\varphi\) is not a great circle; its radius shrinks with the cosine of latitude.
The radius of a parallel at latitude \(\varphi\) is \(R\cos\varphi\), so the arc length for a longitude change \(\Delta\lambda\) (radians) is:
\[\boxed{d_{\lambda} = R\cos\varphi\,|\Delta\lambda|}\]In decimal degrees:
\[d_{\lambda} = R\cos\varphi \cdot \frac{\pi}{180}\,|\Delta\lambda_{\circ}|}\]Key consequence. One degree of longitude spans ≈ 111.1 km at the equator, but shrinks to zero at the poles. At 45° latitude it is ≈ 78.6 km; at 60° ≈ 55.6 km.
Longitude difference → metres
Uses d = 6 371 000 × cos(φ) × |Δλ| × π/180
3 — General Two-Point Distance: Haversine Formula
For two arbitrary positions \((\varphi_1, \lambda_1)\) and \((\varphi_2, \lambda_2)\), the Haversine formula computes the great-circle distance on a sphere exactly (within the spherical assumption):
\[a = \sin^2\!\left(\frac{\Delta\varphi}{2}\right) + \cos\varphi_1\,\cos\varphi_2\,\sin^2\!\left(\frac{\Delta\lambda}{2}\right)\] \[\boxed{d = 2R\,\arctan2\!\left(\sqrt{a},\,\sqrt{1-a}\right)}\]where all angles are in radians. The use of \(\arctan2\) rather than \(\arcsin\) avoids numerical instability for antipodal points.
Why Haversine? The naïve law of cosines form \(d = R\arccos(\sin\varphi_1\sin\varphi_2 + \cos\varphi_1\cos\varphi_2\cos\Delta\lambda)\) suffers from catastrophic cancellation for small separations in floating-point arithmetic. Haversine remains numerically stable at all scales.
Haversine: two-point great-circle distance
Paris → London by default. Uses Haversine with R = 6 371 000 m.
Summary
| Case | Formula | Notes |
|---|---|---|
| Latitude diff only | \(d = R\,|\Delta\varphi|\) | ~111.1 km / degree, latitude-independent |
| Longitude diff only | \(d = R\cos\varphi\,|\Delta\lambda|\) | scales with \(\cos\varphi\); zero at poles |
| General | Haversine | Numerically stable great-circle distance |
All three converge for small \(\Delta\varphi, \Delta\lambda\) — the first two are simply the component projections of the Haversine result in the limit of small separation.