How far can you see across the sea, and at what range will a lighthouse first peek over the curve of the Earth? This calculator answers both, separating the purely geometric horizon from the refraction-corrected visible and radar horizons that mariners actually use.
How it works
Each horizon distance is proportional to the square root of the observer’s eye height, with a constant that bundles the Earth’s radius and a refraction coefficient:
geometric horizon (NM) ≈ 1.93 × √(height in m)
visible horizon (NM) ≈ 2.08 × √(height in m)
radar horizon (NM) ≈ 2.23 × √(height in m)
geographic range to target = visible(eye height) + visible(target height)Radio waves used by radar refract more strongly than light, which is why the radar horizon constant is larger. The geographic range to an object is the sum of two horizon distances because the line of sight grazes the same horizon from both ends.
Example and notes
From a bridge 20 m above the water, the visible horizon lies about 9.3 NM away. A lighthouse with a focal plane 40 m high has its own horizon at about 13.2 NM, so the light should first dip above the horizon at roughly 22.5 NM combined, assuming the light is bright enough to be seen that far. Remember that the actual range a charted light is visible is the smaller of this geographic range and the luminous range printed on the chart.
Practical guidance for mariners and coastal observers
When to use each horizon type
The geometric horizon is a theoretical baseline — it ignores the bending of light entirely. It is rarely used in practical navigation. The visible horizon is what the eye actually sees and is the value used in most chart-work, pilotage, and light-identification problems. The radar horizon is slightly farther because radio waves refract more strongly through the troposphere. It tells you the maximum theoretical detection range for a surface target — in practice, sea clutter, target size, and antenna height all reduce real-world radar detection to something shorter.
Geographic range of lights
When identifying lights at sea, the light’s charted nominal range (its luminous range in good visibility) is printed in the chart. But you cannot see a light that has not yet crested the horizon, regardless of how bright it is. The geographic range — the sum of the observer’s horizon distance and the light’s horizon distance — sets the hard geometric limit. The light becomes visible at the lesser of the two ranges. For a powerful lighthouse with a very high focal plane, the geographic range often exceeds the luminous range and the latter becomes the limit; for a weak harbour light on a low pole, the luminous range is usually the bottleneck.
Dipping distance — a useful pilotage check
When a light is just rising or dipping on the horizon, the combined geographic range tells you your distance off. If a lighthouse (focal plane 35 m) is just dipping below the horizon from your bridge at 15 m height: visible horizon for eye = 2.08 × √15 ≈ 8.1 NM; for the light = 2.08 × √35 ≈ 12.3 NM; combined distance off ≈ 20.4 NM. This dipping-distance technique is a classic coastal navigation fix that requires no other equipment.
Effect of tidal height
These calculations assume the observer’s eye height and the object’s focal-plane height are measured above mean sea level. In practice, the observer’s eye height above the chart datum is what matters. High water adds to both heights; low water can reduce an object’s effective height if it is close to the waterline. For most purposes the variation is small, but it matters for precise distance-off calculations on a large tidal range coast.
Metric and imperial equivalents
The constants in the formula are tuned for heights in metres and ranges in nautical miles, which is the standard in marine navigation. If you have heights in feet, divide by 3.281 before entering them. Radar detection of coastlines and larger targets typically extends well beyond the radar horizon due to reflections and ducting, so treat the radar horizon as a lower bound on detection range, not an upper bound.