NauticTools2Coastal navigation
Geographical range – how far away is the light?
When a lighthouse first appears above the horizon on your approach to the coast, it is rising. At that very moment the distance to the light follows from just two numbers: your height of eye and the height of the light – no radar, no second bearing.
- Height of eye
- Height of light
- Distance to horizon
- Distance in nm
- Full working
What does “rising” and “dipping” mean?
Because the earth is curved, a distant light is hidden below the visible horizon. As you close the coast, the light eventually appears just above the horizon for the first time – it is rising. Sailing away, it sinks below the horizon again and disappears – it is dipping. The distance at that moment is the geographical range of the light for your height of eye.
This distance is made up of two parts that meet exactly at the horizon: from your eye to the horizon, and from the horizon to the light.
When do you need it?
- Fixing your position at night: Take a bearing of the light the moment it rises and you have a bearing and a distance from a known point – a fix without a second object.
- Planning a landfall: Estimate in advance when and at what distance a light should come into view. If it stays out of sight longer than expected, your dead reckoning position is probably off.
Formula and working
- D – distance to the light in nautical miles (nm)
- h – height of eye in metres: the height of your eye above the water
- H – height of the light in metres, from the chart or the list of lights
The calculation takes three steps – exactly as the working in NauticTools2 shows it:
Distance to the horizon
How far does your own horizon reach? 2.075 · √h
Range of the light
From how far away is the light visible above the horizon? 2.075 · √H
Geographical range
Add both parts: D = distance to horizon + range of the light
Why 2.075? Pure geometry would give a factor of about 1.93. Light is bent slightly towards the earth in the atmosphere (terrestrial refraction), so you can see a little further. Some textbooks round it to 2.08 or 2.1; with heights in feet the factor is about 1.15. The result changes only slightly.
Worked example
From a height of eye of 5 m, a light with a height of 62 m rises on the horizon. How far away is the light?
Distance to horizon = 2.075 · √5 = 2.075 · 2.236 ≈ 4.64 nm
Range of the light = 2.075 · √62 = 2.075 · 7.874 ≈ 16.34 nm
Geographical range = 4.64 + 16.34 ≈ 20.98 nm
How far away is the horizon from a height of eye of 3 m?
Distance to horizon = 2.075 · √3 = 2.075 · 1.732 ≈ 3.59 nm
Things to watch out for
- Luminous range: The calculation only holds if the light is bright enough. If its luminous range (given in nm on the chart) is shorter than the calculated distance, the light only becomes visible within that range – it does not rise on the horizon.
- Loom: With powerful lights you often see their glow in the sky or on the clouds well before. What counts is the moment the light itself appears above the horizon.
- Be honest about height of eye: On a sailing yacht it is usually 2 to 3 m in the cockpit. Stand up or climb onto the coachroof and you see further.
- An approximation: Refraction varies with the weather, so the result is a good guide rather than an exact distance.
How it works in NauticTools2
Enter the heights
Height of eye and height of the light in metres.
Calculate
NauticTools2 shows the distance to the light and the distance to the horizon, with the full working of all three steps below.
Save and print
Inputs can be saved under a name of your choice and loaded again later, and the result can be printed as a calculation record – handy for exercises.
Frequently asked questions
How do I work out just the distance to the horizon?
With 2.075 · √height of eye. From 2 m that is about 2.9 nm, from 10 m about 6.6 nm.
What is the difference between luminous range and geographical range?
The luminous range depends on the brightness of the light and is printed on the chart. The geographical range – what is calculated here – depends only on the heights and the curvature of the earth. You see the light only when you are within both.
And if the light is already well above the horizon?
Then this formula no longer helps. You can find the distance instead with a vertical sextant angle: measure the angle between the waterline and the light.
Related tools
- Distance by Vertical Angle – distance to an object of known height from the sextant angle
- Bearing Conversion – convert the bearing of the light to a true bearing
- Distance, Speed and Time – when will you reach the light at your current speed?
NauticTools2 is free and runs on Windows 10 and 11. An Android version is in preparation.
Get it from the Microsoft Store