Distance to the horizon
How far you can see from a height: the horizon distance with no refraction, with optical refraction, and for radio (4/3 Earth), with the rules of thumb and their errors.
Planning and education aid. Not for primary navigation. Full disclaimer
From that height the visible horizon is 38.27 km away (35.7 km with no refraction, 41.22 km for radio).
- This assumes a smooth Earth: terrain, buildings, and trees can block the view or the signal.
- Geometric horizon
- Radio horizon
- Visual horizon, straight line
- Rule of thumb 3.57√h
- Rule of thumb 1.17√h
- Rule of thumb 1.23√h
Provenance
- Computed by
- navigation.los.horizon 1.0.0, core 0.1.0
- Model
- Spherical Earth, R = 6371000 m, refraction k = 0.13 (effective radius 7322989 m)
- Accuracy
- Exact for the model; real refraction varies with the weather, often by 10% or more near the surface
- Notes
- 1 shown with the answer
- Cites
- National Geospatial-Intelligence Agency, Pub. No. 9, The American Practical Navigator (Bowditch)
Something look off?
How we got thisFormula, worked example, sources, and proof
Model: Spherical Earth with effective radius R/(1 − k)
Show your work
Effective Earth radius
Re = R / (1 − k), with k the refraction coefficient6,371 km / (1 − 0.13)= 7,323 kmDistance to the horizon
7,323 km × acos(7,322,989 / (7,322,989 + 100 m))= 38.27 km
The same steps an agent gets from the MCP server with explain: true.
Accuracy: Exact for the model; real refraction varies with the weather, often by 10% or more near the surface
When to use this: Use this when the question is how far away something can be seen or reached from a height: the horizon from a tower or an aircraft, the range at which a light or a landmark comes into view, or the radio horizon for a line-of-sight link, which is longer because the atmosphere bends the signal. The familiar rules of thumb are shown with their errors.
Limitations: It is a smooth sphere with a refraction factor: no terrain, no buildings, and no obstacle between you and the horizon. Real refraction changes with the weather and can move the answer by ten percent or more near the surface, and ducting can carry a signal far past it. For a specific obstacle or a terrain profile, use the visibility and profile tools.
Worked example: From 100 m up. Source: navigation line-of-sight scenario: geometric 35.70 km, optical (k = 0.13) 38.27 km, radio 41.22 km. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.
You enter
- Height above the surface
- 100 m
You get
- Visual horizon
- 38.27 km
- Geometric horizon
- 35.7 km
- Radio horizon
- 41.22 km
- Visual horizon, straight line
- 38.27 km
- Rule of thumb 3.57√h
- 35.7 km
- Rule of thumb 1.17√h
- 21.19 NM
- Rule of thumb 1.23√h
- 22.28 NM
- Visual rule error
- 0.98 km
Review: Not yet independently reviewed by a geodesist.
Last verified: 2026-09-19, 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.
Changes
- 2026-09-19, changed: Distance to the horizon is stable. Its rule-of-thumb figure reproduces all 126 heights of Bowditch's printed Distance of the Horizon table (NGA Pub. 9, Volume II, 2024 edition), in both the nautical and statute columns, to the tenth of a mile the table shows. One row is a misprint: the table gives 29.5 nautical miles at 640 feet where 1.17 times the square root of 640 is 29.6, and that row's own statute figure follows the rule. The tool gives 29.6, and a test pins the difference. Results are unchanged. Changelog
Checked against: 29 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.
Sources
- The American Practical Navigator (Bowditch), National Geospatial-Intelligence Agency, Pub. No. 9, 2024 edition. Volume II, Table 12 (Distance of the Horizon) and Table 14 (Dip of the Sea Short of the Horizon).