Radii of curvature and degree lengths
At a latitude: the meridional and prime-vertical radii of curvature, their Gaussian mean, the radius in any azimuth, the length of one degree of latitude and longitude, and the meridian arc from the equator or between two latitudes.
At this latitude one degree of latitude is 111,131.778 m and one degree of longitude is 78,846.835 m.
- One degree of longitude
- Meridional radius M
- Prime-vertical radius N
- Gaussian mean radius √(MN)
- Meridian arc from the equator
Provenance
- Computed by
- geodesy.ellipsoid.radii 1.0.0, core 0.1.0
- Model
- Radii of curvature and meridian arc, WGS 84
- Accuracy
- Meridian arcs within 1 nm of GeographicLib's geodesic along the meridian; radii exact in double precision
- Notes
- None
- Cites
- Snyder, J. P., U.S. Geological Survey, Map Projections: A Working Manual, USGS Professional Paper 1395; Karney, C. F. F., GeographicLib, GeographicLib Geocentric and LocalCartesian classes
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Closed-form radii; meridian arc a[E(φ, e) − e² sinφ cosφ/W] by Carlson's elliptic integrals
Accuracy: Meridian arcs within 1 nm of GeographicLib's geodesic along the meridian; radii exact in double precision
When to use this: Use this when a local calculation needs the curvature of the Earth where you are: how far a degree of latitude and a degree of longitude run on the ground, the radius a survey reduction or a projection wants at that latitude, the radius in a particular azimuth, and the meridian arc from the equator. It is the ellipsoid seen from one place rather than as a whole.
Limitations: These are curvatures at a point, not distances between places: a degree of longitude is the parallel through the point, which is not the shortest way between its ends, so a route calculation wants a geodesic and not this. The meridional and prime-vertical radii differ by about 21 km at the equator and meet only at the poles, so a single radius of the Earth is always an answer to a narrower question than it looks. Above 89 degrees a degree of longitude is a few dozen metres and the figures lose their usefulness before they lose their accuracy.
Worked example: 45° N on WGS 84. Source: add-geodesy-suite scenario: one degree of latitude ≈ 111,131.78 m, one degree of longitude ≈ 78,846.84 m. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Latitude
- 45 deg
You get
- One degree of latitude
- 111,131.778 m
- One degree of longitude
- 78,846.835 m
- Meridional radius M
- 6,367,381.816 m
- Prime-vertical radius N
- 6,388,838.29 m
- Gaussian mean radius √(MN)
- 6,378,101.03 m
- Meridian arc from the equator
- 4,984,944.378 m
Review: Not yet independently reviewed by a geodesist.
Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.
Status: version 1.0.0, core 0.1.0. See this tool in the verification report.
Checked against: 25 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- Map Projections: A Working Manual, USGS Professional Paper 1395, Snyder, J. P., U.S. Geological Survey, Professional Paper 1395. Chapter 3: auxiliary latitudes, radii of curvature, and meridian distance.
- GeographicLib Geocentric and LocalCartesian classes, Karney, C. F. F., GeographicLib, GeographicLib 2.x. Geocentric.cpp: Vermeille's closed-form inverse; LocalCartesian.cpp: ENU rotation. Read free.