Distance to Horizon Calculator
Enter your eye height and get how far away the horizon is, with and without standard atmospheric refraction, plus how far you can see a tall object and how much of it the curve hides.
Measure from the water or ground surface to your eyes, including the deck, cliff or floor you stand on. The result updates as you type.
Horizon from an eye height of 5.7 ft
3.16 mi
with normal refraction · 2.92 mi geometric (no atmosphere)
| With standard refraction | 3.16 mi · 5.08 km · 2.74 nmi |
| Geometric (no refraction) | 2.92 mi · 4.71 km · 2.54 nmi |
Refraction bends light slightly around the curve, so you normally see a little farther than the geometry alone allows. How much varies with the weather: over cold water in spring the bending can be far stronger, and over hot ground it can reverse (mirages). The refraction figure uses standard conditions; treat both numbers as the range of the usual case, not a limit.
How to work out the distance to the horizon
- Enter your eye height. Height of your eyes above the water or flat ground, including any deck, cliff or building.
- Read the horizon distance. With normal atmospheric refraction, and the pure geometric figure.
- Add a distant object (optional). Its height gives the farthest distance at which its top can still be seen.
- Add a distance (optional). Shows how much of an object at that distance is hidden below the horizon.
10 reasons to calculate the distance to the horizon
Spotting a lighthouse from a boat
How far out its light first appears over the horizon from your deck height.
Choosing a lookout height
How much farther you see from a 30 ft tower than from the beach.
Ham radio and VHF range
Line-of-sight range between two antennas starts from both horizon distances.
Photographing a distant skyline
How much of the buildings will be hidden behind the curve across a lake.
Drone pilots keeping visual line of sight
How far a low aircraft can be before the horizon starts to hide it.
Sailing in poor conditions
When another vessel of known height should come into view.
Answering a curvature question
Work through how much of a far shore should be hidden, using the formula shown.
Hiking viewpoints
How far a summit view reaches over flat ground or the sea.
Search and rescue planning
Rough visibility range of a lookout over open water.
Physics and geometry teaching
The Pythagorean derivation, with real numbers.
The formulas used
For an observer at height h above a smooth spherical Earth of radius R = 6,371 km, the straight-line distance to the geometric horizon is d = √(2Rh + h²), from Pythagoras (Wikipedia, 'Horizon'). For everyday heights this is about 3.57 km × √h with h in metres.
Air normally bends light slightly downward, so you see a little farther. A. T. Young (San Diego State University) describes the usual allowance: replace R with an effective radius R′ = 7R/6, which gives about 3.86 km × √h. Both results are shown. The farthest distance at which a distant object's top is visible is the sum of the two horizon distances (yours and the object's). The hidden height of an object at a given distance is how far below your grazing line of sight its base sits beyond your horizon, worked with the same model.
Limits: refraction depends strongly on the temperature of the air near the surface. Over cold water in spring it can let you see far beyond these figures; over hot ground it can reduce them and create mirages. Waves, haze and terrain are not modelled. Use the figures as the standard-conditions estimate, not as a guarantee of what you will see.
Frequently asked questions
How far away is the horizon?
For eyes about 1.7 m (5.6 ft) above the sea, roughly 4.7 km (2.9 mi) geometrically and about 5 km (3.1 mi) with normal refraction.
What is the formula for distance to the horizon?
d = √(2Rh + h²), where R is the Earth's radius and h your eye height. For small heights, d ≈ 3.57√h km (h in metres), or about 3.86√h km with normal refraction.
Does refraction let you see farther?
Usually, yes. Air bends light slightly around the curve; the standard allowance is to use 7/6 of the Earth's radius.
How far can I see a lighthouse?
Enter your eye height and the lighthouse's height. The farthest distance at which its top is visible is your horizon distance plus its horizon distance.
How much of a distant object is hidden by the curve?
Enter the object's distance. The page shows the hidden height for the geometric and refracted cases.
Does this work for mountains or aircraft?
Yes for height above a smooth surface. It does not know about the terrain between you and the target.
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