Distance to the horizon and geographic range of a light
At sea, the distance to the horizon in nautical miles is about 2.08 × √h, with h your height of eye in metres. A light comes into sight when its horizon meets yours, so its geographic range is about 2.08 × (√h eye + √h light). From a height of eye of 2 m the horizon is about 2.9 NM away; a light 40 m high can be seen from about 16.1 NM, if it is bright enough.
Distance to the horizon and light range calculator
Enter heights above sea level. Leave the light height at 0 to get only your own horizon.
- Your horizon
- 2.9 NM · 5.4 km · 3.4 mi
- Light's horizon
- 13.2 NM · 24.4 km · 15.1 mi
- Geographic range
- 16.1 NM · 29.8 km · 18.5 mi
d = 2.08 × (√h eye + √h light), heights in metres, d in nautical miles
Worked examples
1. Horizon from the deck
Height of eye 3 m: 2.08 × √3 = 2.08 × 1.73.
About 3.6 NM (6.7 km, 4.1 statute miles).
2. When will the light appear?
A light 40 m above sea level, charted range 15 M; your eye is 2 m above the water. Geographic range: 2.08 × (√2 + √40) = 2.08 × (1.41 + 6.32).
Geographic range about 16.1 NM. In a visibility of about 10 NM its luminous range is its 15 NM nominal range, so you first see it at about 15 NM: here the light's intensity, not the horizon, is the limit.
3. A light rising on the horizon
Approaching at night, you see a 30 m light just appear on the horizon from a 2 m height of eye. 2.08 × (√2 + √30) = 2.08 × (1.41 + 5.48).
You are about 14.3 NM from the light. With its bearing, that gives a rough position. Check it: refraction can change the real distance.
4. Heights in feet
Eye 9 ft, light 120 ft. Convert: 2.74 m and 36.58 m, then 2.08 × (1.66 + 6.05). Bowditch's feet formula, 1.17 × (√9 + √120), gives 16.3 NM.
About 16.0 NM; the two formulas differ by about 2 %, much less than the effect of real refraction.
How the formula works
The Earth curves away from you, so from a height h you can see only as far as the point where your line of sight touches the sea. For a sphere of mean radius 6,371 km that distance is √(2Rh), about 1.93 × √h nautical miles with h in metres. The atmosphere bends light slightly downwards, which in standard conditions adds about 8 % and gives the navigator's rule: horizon ≈ 2.08 × √h.
A light has its own horizon too. You can see it when your horizon and the light's horizon meet, so its geographic range is the sum of the two: 2.08 × (√h eye + √h light). Light heights on charts and in lists of lights are given above a sea-level datum; check the chart's height datum and add your own height of eye above the water.
Navigation tables use slightly different coefficients, usually between 2.03 and 2.12, because refraction is only an average. Bowditch's Table 12 is equivalent to 1.17 × √h with h in feet (about 2.12 with metres). The differences are a few per cent; abnormal refraction over cold water or in a heat haze can change the real horizon far more.
Geographic range is only half the answer. The luminous range is the furthest distance at which a light can be seen in the current visibility, set by its intensity. The nominal range is the luminous range in a meteorological visibility of 10 NM; it is the range usually printed on charts and in light lists. You first see a light at the lesser of its geographic and luminous ranges.
Watching a light rise above, or dip below, the horizon gives a distance-off. With the bearing of the same light, that is a quick position check at night. Treat it as approximate: wave height, your exact height of eye and refraction all change the result.
Frequently asked questions
How far is the horizon at sea?
About 2.08 × √h nautical miles, with h your height of eye in metres. At 2 m it is about 2.9 NM; at 10 m about 6.6 NM; at 30 m about 11.4 NM.
How do I calculate the geographic range of a light?
Add the light's horizon to yours: 2.08 × (√h eye + √h light), heights in metres and the result in nautical miles. A 40 m light seen from 2 m gives about 16.1 NM.
What is the difference between geographic, luminous and nominal range?
Geographic range depends on the heights of the light and the observer. Luminous range depends on the light's intensity and the visibility. Nominal range is the luminous range in 10 NM visibility. You see the light at the lesser of the geographic and luminous ranges.
Why do some books use 2.03 or 2.12 instead of 2.08?
They assume a slightly different average refraction. Bowditch's 1.17 × √h in feet is about 2.12 × √h in metres. The results differ by a few per cent, which is smaller than the uncertainty in real refraction.
Can I use this formula for the radar horizon?
No. Radar waves bend more than light in standard conditions, so the radar horizon is further. Use the radar formula from your radar manual or navigation text.
Practise lights for your exam
Light characteristics and ranges are part of coastal navigation exams. Practise identifying vessel lights and buoyage, then work a night-passage plan with real charted lights.
Sources
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