Horizon Distance Calculator
Enter how high your eyes are above the surface to find how far away the horizon is. Add a target height, such as a lighthouse or a ship's mast, to get the greatest distance at which you can still see it, or a target distance to see how much of it is hidden by the curve. Works with or without atmospheric refraction, on Earth or other worlds.
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Height of your eyes above the surface (sea level), e.g. 1.7 m standing on a beach
Top of a lighthouse, ship mast, mountain or building, same unit
Shows how much of the target is hidden below the horizon
How to use the Horizon Distance Calculator
- Enter your eye height above the surface or sea level and choose the unit.
- Optionally enter a target height (and its distance) to check whether you can see it.
- Pick the planet or moon and the refraction setting.
- Choose the distance unit and read the horizon distance and visibility range.
- Use the tables to compare other heights and other worlds.
Horizon distance formulas
- Surface (arc) distance: d = R × arccos(R / (R + h)). This is the distance you would travel along the ground or sea to reach the horizon.
- Line of sight: dlos = √(h(2R + h)), the straight line from your eye to the horizon point (from the Pythagorean theorem).
- Low-height approximation: d ≈ √(2Rh). On Earth: d ≈ 3.57√h km (h in m) or d ≈ 1.22√h mi (h in ft) geometrically; with standard refraction d ≈ 3.83√h km or 1.31√h mi (1.14√h nautical miles).
- Refraction: replace R with the effective radius R / (1 − k). Typical daytime light uses k ≈ 0.13; radio and microwave links use k = 0.25 (the “4/3 Earth”).
- Visibility of a target: dmax = d(observer) + d(target). The hidden height of an object at distance D is R′ / cos((D − dobs) / R′) − R′.
Horizon distance on Earth
| Eye height | Example | Geometric | With refraction (k = 0.13) |
|---|---|---|---|
| 1.7 m (5.6 ft) | Standing at the water's edge | 4.7 km (2.9 mi) | 5.0 km (3.1 mi) |
| 10 m (33 ft) | Third-floor window, small ship bridge | 11.3 km (7.0 mi) | 12.1 km (7.5 mi) |
| 100 m (328 ft) | Coastal cliff, 30-story building | 35.7 km (22.2 mi) | 38.3 km (23.8 mi) |
| 1,000 m (3,281 ft) | Mountain summit above the sea | 113 km (70 mi) | 121 km (75 mi) |
| 10,668 m (35,000 ft) | Airliner cruising altitude | 368 km (229 mi) | 395 km (245 mi) |
Good to know
- Height matters less than you think. Because distance grows with the square root of height, climbing four times higher only doubles the horizon distance.
- Measure height above the surface you look across: above sea level for the ocean, or above the plain for flat land. Hills and waves in between can block the view earlier.
- Refraction varies. Over cold water on a warm day, strong inversions can lift distant ships and coasts into view (looming) or create mirages; over hot ground the horizon can appear closer.
- Navigation lights: the “geographic range” on nautical charts is calculated the same way, using the light's height and a standard observer eye height.
Frequently asked questions
How far away is the horizon?
For a person standing at the edge of the sea with eyes about 1.7 m (5.6 ft) up, the geometric horizon is about 4.7 km (2.9 mi) away; normal atmospheric refraction stretches it to about 5.0 km (3.1 mi). From 100 m up it is about 36 km geometrically, and from an airliner at 35,000 ft about 370 km (230 mi).
What is the formula for the distance to the horizon?
The exact surface distance is d = R × arccos(R / (R + h)), and the straight line of sight is √(2Rh + h²), where R is the planet's radius and h your height. For heights much smaller than R both reduce to d ≈ √(2Rh). On Earth that gives d ≈ 3.57 × √h km with h in meters, or d ≈ 1.22 × √h miles with h in feet.
How does refraction change the horizon distance?
Air density decreases with height, so light rays bend slightly downward and follow the Earth's curve. This is modeled by using a larger effective radius R / (1 − k). With the typical coefficient k = 0.13 the horizon is about 7% farther (≈ 3.83 × √h km). Radio engineers use the 4/3-Earth model (k = 0.25), and unusual temperature inversions can extend or shrink the visible horizon a lot.
How far can I see a lighthouse or a ship?
Add the observer's horizon distance to the target's horizon distance. A lighthouse 50 m tall seen from a deck 25 m up is visible to about 27.1 + 19.1 = 46.2 km (about 25 nautical miles) with standard refraction. Beyond that the light is below the horizon, although its glow on clouds may still be seen.
How much of a distant object is hidden by the Earth's curve?
First subtract your own horizon distance from the distance to the object. The hidden height is then about R′ / cos(x / R′) − R′, where x is that remaining distance and R′ the effective radius. From an eye height of 2 m with standard refraction, about 14.5 m of a ship 20 km away is hidden.
How far is the horizon on the Moon or Mars?
Smaller worlds curve away faster. At an eye height of 1.7 m the horizon is about 2.4 km away on the Moon (radius 1,737 km) and 3.4 km on Mars (3,390 km), compared with 4.7 km on Earth without refraction.
Is the horizon distance the same as how far I can see?
No. It is the distance to where the surface meets the sky. Taller things beyond it, such as mountains, clouds and the Moon, stay visible, and haze, dust or fog often limit visibility to much less than the horizon distance.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.