How far can you see? Horizon distance calculator
From eye level on a beach, about 1.7 m above the water, you can see roughly 5 km. The distance to the horizon grows with the square root of your height, about 3.86·√h kilometres for h in metres once normal refraction is counted. The calculator also shows how much of a distant ship or peak the curve hides.
Eye level above the sea or plain you are looking across. On a beach that is about 1.7 m.
A ship’s mast, a lighthouse, a mountain. Leave it at 0 for something at surface level.
Optional. Used only to work out how much of the object the curve hides.
Air bends light downward, which pushes the horizon a little farther out.
You stand on top of the bulge, so the surface falls away from your own feet. The green line is your line of sight: it leaves the eye at h₁, grazes the surface at the amber horizon point and keeps dropping. Everything in the shaded wedge past that point is out of sight, which is why the lower part of the object is drawn in amber. The dashed outline farther out is the same object at its limit, where the top only grazes the ray. That distance is d₁ + d₂, your own horizon plus the object’s own horizon, and the corner note prints it. Distances are to scale, and the second corner note gives the factor the heights are stretched by.
Smooth sphere, one refraction factor for the whole path, clear air. Real ground gets in the way long before geometry does, and haze usually limits what you can actually make out well before the horizon.
How it works
- 01
Measure your height above the surface
What counts is how high your eyes sit above the ground or water you are looking across, not your altitude above sea level. Standing on a beach that is about 1.7 m. Standing on a 1 000 m ridge above the sea it is 1 000 m.
- 02
Find your own horizon
The sight line leaves your eye and grazes the surface at d1 = 2·k·R·h1. With R = 6 371 km that is 3.57·h km for h in metres, stretched to about 3.86·h once air bends the light. From 1.7 m the horizon is 5.03 km away.
- 03
Add the object’s own horizon
A tall object sticks up over the curve, so it is visible from farther than its base. Its own horizon d2 follows the same formula, and the top stays in sight out to d1 + d2. A 30 m mast adds 21.1 km to your 5.03 km.
- 04
Work out how much is hidden
Past your horizon, the surface keeps dropping below the sight line: hhidden = (D − d1)2 / (2·k·R). At 15 km the drop is 6.7 m, so the bottom 6.7 m of that ship is out of sight while the mast still shows.
Formulas
- d — distance to the horizon, km
- h — height above the surface, m
- R — earth radius, 6 371 km
- k — refraction factor (1 = none, ≈ 1.17 = normal air; the 3.86 shorthand assumes k = 1.17, use 3.57 for k = 1)
- h1 — your height above the surface, m
- h2 — height of the object, m
- the two horizons meet at the same grazing point
- D — distance to the object, km (zero hidden while D ≤ d1)
- d1 — your own horizon, km
- visible part = h2 − hhidden
Worked example
- your horizon: d1 = 3.86·1.7 = 5.03 km (4.65 km with no refraction)
- mast’s horizon: d2 = 3.86·30 = 21.1 km
- the masthead stays in sight to d1 + d2 = 26.2 km (16.3 mi, 14.1 nmi)
- at 15 km: hhidden = (15 − 5.03)2 / (2·1.17·6371) = 6.7 m
- → the lowest 6.7 m of the ship is below the horizon, 23.3 m of mast still shows
FAQ
- How far can you see from a mountain of 1 000 m or 3 000 m?
- From 1 000 m the horizon is about 122 km away (76 mi), from 3 000 m about 211 km (131 mi). Without refraction the same heights give 113 km and 196 km. Distant peaks show up much farther still, because their own height is added: a 3 000 m summit seen from a 1 000 m ridge can appear at over 330 km.
- Why do ships disappear hull first?
- Because the sea curves away below your line of sight. Past your horizon the surface keeps dropping, so the hull goes out of sight while the masts are still above the line. Watching that happen, and seeing the ship reappear top-first as it returns, is the oldest everyday proof that the Earth is round.
- How much does atmospheric refraction change the answer?
- Air is denser near the surface, so light travelling almost horizontally bends gently downward and follows the curve a little way. Under normal conditions that pushes the horizon about 8% farther out, which is the difference between 3.57·√h and 3.86·√h. It is not a constant. Over cold water under a warm layer, refraction gets much stronger, objects appear stretched upward, and coastlines can loom into view hundreds of kilometres away.
- Why is the radio horizon farther than the visible one?
- Radio waves bend more strongly than light in the same air. Link planning uses an effective earth radius of k = 4/3 instead of k ≈ 1.17, which turns 3.86·√h into 4.12·√h, about 7% farther. For antenna heights and line-of-sight range, use the radio horizon calculator listed under related tools below.
- Can you see the curvature of the Earth from an airliner?
- Barely. At a cruising altitude of 11 km the horizon is about 405 km away and sits roughly 3° below eye level, but the horizon line itself is still an extremely shallow arc. Most photos that appear to show a strong curve are made by a wide-angle lens or by the curved cabin window. The bend becomes genuinely obvious only from far higher, around 30 km and above.
This is the view over a perfectly smooth Earth. The real one has hills in the way. Waveshed draws what is visible from a point over actual elevation data, free in your browser.
See your real view on the map →Related guides
- Heights: AGL vs AMSL — the formula wants your height above the surface you are looking across, not your altitude on the map
- Elevation profile & measuring tools — the simulator’s side view shows where a real hill, not the horizon, cuts the view short
Sources & further reading
- Bowditch, The American Practical Navigator (NGA Pub. No. 9) ↗ — the standard navigational treatment of the visible horizon, dip and geographic range
- Distance to the Horizon — Andrew T. Young, San Diego State University ↗ — terrestrial refraction, the R′ = 7/6·R shortcut behind the 3.86 coefficient, and when it fails
- Dip of the Horizon — Andrew T. Young, San Diego State University ↗ — geometry of the grazing ray and how far the horizon drops below eye level
- Horizon — Wikipedia ↗ — derivation of √(2·R·h) and the 3.57·√h and 3.86·√h forms
Related tools
Radio horizon calculator
How far a signal reaches over smooth earth, from antenna height.
Earth bulge calculator
Height the Earth’s curvature adds mid-path, with k-factor refraction.
Distance & bearing calculator
Great-circle distance, bearing and elevation angle between two coordinates, summit to summit.
Horizon mask calculator (ridge elevation angle)
The elevation angle a ridge blocks, and whether GPS, satellite messengers or a GEO satellite still clear it.