The total visibility range is the distance at which an observer at the given height first sees the top of an object of the given height rising above the geometric horizon.
Result
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Facts and limits of this method
| Formula | range = k × (√h_observer + √h_object), where k = 3.57 without refraction or 3.86 with refraction |
|---|---|
| Example | observer 1.7 m + lighthouse 20 m (with refraction) ≈ 5.03 + 17.26 = 22.29 km |
| Earth's radius | Taken as 6371 km (mean radius) |
When it misleads you
- The formula assumes a perfectly smooth spherical Earth with no mountains or other obstructions between the observer and the object — the real visibility range on terrain with relief may differ in either direction.
- Atmospheric refraction (the bending of light rays) extends the visibility range by about 8% on average; the 3.86 coefficient is a common approximation for standard conditions, while actual refraction depends on air temperature and pressure.
- The calculation doesn't account for atmospheric clarity (haze, fog, precipitation) — even a geometrically visible object may not be discernible to the naked eye under poor visibility.
- "Object height" means the height of its topmost point above sea level — for a lighthouse this is usually the height of the light itself; for a ship, the height of its tallest superstructure or mast.
How it is calculated
The entered observer and object heights are converted to meters (multiplied by 0.3048 if feet is selected).
For each of the two heights separately, the distance to the horizon is computed using d = k × √h, where k is the coefficient 3.57 (without refraction) or 3.86 (with refraction).
The total visibility range equals the sum of these two distances — this is the standard maritime and surveying approach used for calculating a lighthouse's or landmark's nominal visibility range.
The result in kilometers is additionally converted to miles, and each height's contribution is shown separately for clarity.
Questions and answers
Why is the range a sum instead of just the distance to the object's horizon?
Because both the observer and the object are elevated above Earth's surface, and each height pushes the line of sight out by its own amount — the total range is the observer's contribution plus the object's contribution.
Where is this kind of calculation used?
In maritime navigation for calculating a lighthouse's nominal visibility range, in surveying, in line-of-sight radio communication, and when planning observation towers or lookout posts.
What if the object height is zero?
Then the calculation reduces to the ordinary horizon distance from the observer's height alone — the object's contribution is zero in that case.
Is my data sent anywhere?
No, the whole calculation runs in your browser.