Distance to Horizon Calculator
Earth curves away from any observer, so the horizon sits a finite distance ahead. This tool uses a smooth-sphere model (R = 6,371 km) and returns both the geometric horizon distance and the refraction-corrected one in kilometres, miles and nautical miles.
Enter a positive height (max 100 km — above that you should be using a satellite-footprint calculator).
Geometric horizon distance
4.65 km
—
Pure geometry d = √(2·R·h + h²) — ignores atmospheric refraction, mathematically exact for a smooth sphere.
Visible (refraction-corrected) horizon
4.99 km
—
Uses the standard refraction coefficient k = 0.13 (Bowditch, American Practical Navigator). Light bends slightly downward in the atmosphere, so the apparent horizon is ~8 % farther than the geometric one.
Assumes a smooth spherical Earth (R = 6,371 km) and ignores terrain obstruction. Refraction uses an average-atmosphere model; actual visibility varies with temperature gradients.
Formula
d_geom = √(2·R·h + h²); d_ref = √(2·R'·h + h²), R' = R / (1 − k), k = 0.13
- · R is the IUGG mean Earth radius 6,371.0088 km.
- · The geometric expression d = √(2·R·h + h²) comes from the Pythagorean theorem on the right triangle formed by the eye, the centre of the Earth, and the tangent point.
- · For h ≪ R it simplifies to d ≈ √(2·R·h); the common rule of thumb is d_km ≈ 3.57·√h_m.
- · The refraction coefficient k = 0.13 follows Bowditch (American Practical Navigator) and standard surveying practice — equivalent to using an effective Earth radius 7/6·R.
- · Refraction-corrected horizons are about 7–8 % farther: d_km ≈ 3.86·√h_m.
- · Assumes a smooth spherical Earth in standard-atmosphere conditions; terrain, sea swell and anomalous refraction (mirages, looming) are not modelled.
Frequently asked
How far is the horizon when standing on the beach?
A standing adult has an eye height of about 1.6–1.8 m. Plugging into the geometric formula gives d ≈ √(2 · 6,371 km · 0.0017 km) ≈ 4.65 km, plus refraction puts the visible horizon ~5.0 km away. Anything beyond that distance is hidden by the curvature of the Earth.
How far is the horizon from a cruising airliner (10 km / 33,000 ft)?
At a cruising altitude of 10,000 m, the geometric horizon is d = √(2 · 6,371,000 · 10,000 + 10,000²) ≈ 357 km; with refraction it reaches ≈ 381 km. So an airliner over the Mediterranean can in principle see a 700 km diameter circle — though human visual acuity and the window frame limit what you actually pick out.
How do I work out how far away a tall ship or lighthouse first becomes visible?
The maximum distance at which two objects can see each other is the sum of their horizon distances: D = d(h₁) + d(h₂), where h₁ is your eye height and h₂ is the top of the object. For example, your eye at 1.7 m (d ≈ 4.65 km) plus a 30 m mast (d ≈ 19.6 km) means the top of the mast first peeks above the horizon when you are about 24 km apart. Just run the calculator twice and add the two distances.
Geometric vs refraction-corrected — which one should I trust?
Day-to-day, navigation and photography are best served by the refraction-corrected figure — it tracks what your eye actually sees. The pure-geometric number is the spherical-Earth lower bound and is more useful for engineering or line-of-sight obstruction analysis. Be aware: real atmospheric conditions (temperature gradients, fog, hot sea surfaces) push the refraction coefficient k between roughly 0 (very thin air, mirages) and 0.25+ (strong inversions), so real visible range can swing by more than ±10 %.
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