Ellipsoid parameters
The defining and derived parameters of a reference ellipsoid: semi-minor axis, flattening, eccentricities, third flattening, and the mean, authalic, and volumetric radii.
The semi-minor axis is 6,356,752.314 m, with flattening 0.00335281066475.
- Semi-major axis a
- Flattening f
- Inverse flattening 1/f
- First eccentricity squared e²
- Second eccentricity squared e′²
- Third flattening n
Provenance
- Computed by
- geodesy.ellipsoid.parameters 1.0.0, core 0.1.0
- Model
- Ellipsoid relations, WGS 84
- Accuracy
- Exact formulas evaluated in double precision (relative error near 1e-16)
- Notes
- None
- Cites
- National Geospatial-Intelligence Agency, Department of Defense World Geodetic System 1984, NGA.STND.0036; Moritz, H., Journal of Geodesy, Geodetic Reference System 1980; Snyder, J. P., U.S. Geological Survey, Map Projections: A Working Manual, USGS Professional Paper 1395
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How we got thisFormula, worked example, sources, and proof
Model: Closed-form ellipsoid relations; quarter meridian by Carlson's elliptic integrals
Accuracy: Exact formulas evaluated in double precision (relative error near 1e-16)
When to use this: Use this when a calculation needs the shape of the Earth written out: the semi-minor axis, the flattening, the two eccentricities, the third flattening, and the mean, authalic, and volumetric radii of a named ellipsoid, or of one you give by its own two defining numbers. It is what a projection, a datum transformation, or a geodesic calculation is set up from, and what to quote when a result has to say which figure of the Earth it used.
Limitations: An ellipsoid is a figure, not a datum: WGS 84 and GRS 80 differ in the last digits of their flattening and are still different realizations of position, so naming the ellipsoid does not pin down the coordinates. The derived values follow exactly from the two defining ones, so a custom ellipsoid given with a rounded flattening carries that rounding into everything below it. The radii are the ones a sphere would need to match the ellipsoid in one respect each -- mean, equal area, equal volume -- and they are not interchangeable: using the volumetric radius where an area calculation wanted the authalic one is a quiet error of about a part in a million.
Worked example: WGS 84. Source: add-geodesy-suite scenario: a = 6,378,137 m, 1/f = 298.257223563, b = 6,356,752.314245… m. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Ellipsoid
- wgs84
You get
- Semi-minor axis b
- 6,356,752.314 m
- Semi-major axis a
- 6,378,137 m
- Flattening f
- 0.00335281066475
- Inverse flattening 1/f
- 298.257223563
- First eccentricity squared e²
- 0.00669437999014
- Second eccentricity squared e′²
- 0.00673949674228
- Third flattening n
- 0.00167922038638
- Mean radius R1
- 6,371,008.771 m
- Authalic radius R2
- 6,371,007.181 m
- Volumetric radius R3
- 6,371,000.79 m
- Quarter meridian
- 10,001,965.729 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: 23 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- Department of Defense World Geodetic System 1984, NGA.STND.0036, National Geospatial-Intelligence Agency, NGA.STND.0036_1.0.0_WGS84. Table 3.1 (defining parameters) and Table 3.3 (derived geometric constants).
- Geodetic Reference System 1980, Moritz, H., Journal of Geodesy, Journal of Geodesy 74(1). Derived geometric constants and mean radii R1, R2, R3.
- 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.
Terms
- R1: IUGG mean Earth radius
- The radius of a sphere the size of the Earth on average, (2a + b) ÷ 3 = 6,371,008.771 m for WGS 84. Spherical formulas use it when they treat the Earth as a ball. Source: Geodetic Reference System 1980