3D distance between two points

The straight-line distance between two points with heights, such as a drone and its ground station, through Earth-centered coordinates, with the ground distance, height difference, and elevation angle.

Planning and education aid. Not for primary navigation. Full disclaimer

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

2,003.694m

The points are 2,003.694 m apart in a straight line, and 2,000 m apart on the ground.

Ground distance
Height difference
Elevation angle
Azimuth
Flat-Earth distance
Flat-Earth elevation angle

Provenance

Computed by: navigation.vector.distance-3d 1.0.0, core 0.1.0
Model: Both points to WGS 84 ECEF (sea-level heights first become ellipsoidal, h = H + N, with EGM96); straight-line distance = |ΔECEF|; elevation and azimuth from the east-north-up frame at the first point; ground distance by the Karney geodesic inverse
Accuracy: Exact geometry for the heights given. A sea-level height carries the geoid's error, about 0.5-1 m for EGM96.
Notes: None
Cites: Karney, C. F. F., GeographicLib, GeographicLib Geocentric and LocalCartesian classes

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First point latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees First point longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees First point height Like 250 m · plain numbers mean m Second point latitude Decimal degrees, north positive, like 40.6413 · plain numbers mean degrees Second point longitude Decimal degrees, east positive, like -73.7781 · plain numbers mean degrees Second point height Like 370 m · plain numbers mean m

More options 3

First height isoptional hae (above the ellipsoid, like GPS; default) or msl (above sea level) Second height isoptional hae (above the ellipsoid, like GPS; default) or msl (above sea level) Geoid modeloptional none (default) or egm96, to turn sea-level heights into ellipsoidal ones

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How we got thisFormula, worked example, sources, and proof

Model: Both points to WGS 84 ECEF (sea-level heights first become ellipsoidal, h = H + N, with EGM96); straight-line distance = |ΔECEF|; elevation and azimuth from the east-north-up frame at the first point; ground distance by the Karney geodesic inverse

Show your work

Ground distance
geodesic inverse on WGS 84
40°, -105° to 40.018012°, -105° = 2,000 m
Straight-line distance
√(ΔX² + ΔY² + ΔZ²) in Earth-centered coordinates
√(309.021² + 1,153.282² + 1,609.11²) = 2,003.694 m

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Exact geometry for the heights given. A sea-level height carries the geoid's error, about 0.5-1 m for EGM96.

When to use this: Use this for the straight-line distance between two points at different heights -- a drone and its ground station, an aircraft and a transmitter, a summit and a valley floor. It also gives the ground distance, the elevation angle and the azimuth, and the flat-earth answer beside them so you can see what curvature costs.

Limitations: Say which kind of height each is. A GPS height is above the ellipsoid and a map height is above sea level, and they differ by tens of metres; give `msl` and the tool converts through EGM96, which carries about half a metre to a metre of its own. This is geometry, not visibility: it says where the target is, not whether you can see it -- terrain, buildings and the Earth's curve are the look-angles tool next door. And the flat-earth figures are there for comparison, not for use; over a hundred kilometres they are wrong by enough to matter.

Worked example: A ground station at 250 m and a drone 2,000 m north at 370 m. Source: add-navigation-and-geometry drone scenario: straight-line 2,003.694 m, elevation 3.4245°; flat Earth 2,003.597 m and 3.4336°. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

You enter

First point height: 250 m
Second point height: 370 m
First point latitude: 40 deg
Second point latitude: 40.018012369978514 deg
First point longitude: -105 deg
Second point longitude: -105 deg

You get

Straight-line distance: 2,003.694 m
Ground distance: 2,000 m
Height difference: 120 m
Elevation angle: 3.4245°
Azimuth: 0°
Flat-Earth distance: 2,003.597 m
Flat-Earth elevation angle: 3.4336°
Heights used: Both heights above the WGS 84 ellipsoid

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: 20 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

GeographicLib Geocentric and LocalCartesian classes, Karney, C. F. F., GeographicLib, GeographicLib 2.x. Geocentric.cpp: geodetic to ECEF; LocalCartesian.cpp: the east-north-up rotation. Read free.

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