How to use the distance to horizon calculator
- Choose what to find: the distance to the horizon, how far you can see an object of a given height, or how much of an object is hidden.
- Pick the units, then enter your eye height, and the object height or its distance if asked.
- Choose whether to allow for the bending of light, which stretches the horizon a little.
- Read the result in kilometers, miles and nautical miles, and the steps that produced it.
Formula
Distance to the horizon along the surface d = R x acos(R / (R + h))
Shortcut for small h: d is about sqrt(2 R h), or 3.57 sqrt(h in meters) kilometers
Seeing an object of height H: d = d(h) + d(H)
Effective radius with bent light: R / (1 - k)
Height hidden at ground distance D: R x (1 / cos((D - d(h)) / R) - 1)
R is the radius of the Earth, 6,371 km as a mean example, and k is the refraction coefficient, which is 0 for straight light.
A boat deck and a hilltop
With an eye height of 2 meters the horizon is 5.05 km away, which is 3.14 miles or 2.73 nautical miles. A lighthouse 50 meters tall comes into view from 5.05 + 25.24 = 30.29 km away.
Allowing for bent light with the example coefficient 0.13 stretches the 2-meter horizon to 5.41 km. For an eye height of 600 feet the horizon is 29.99 miles. An object 20 km away seen from a 2-meter eye height has 17.54 meters of its base hidden below the horizon.
Why the horizon is where it is
On a smooth sphere, your line of sight to the horizon just grazes the surface. The higher your eye, the further the surface falls away before your line of sight meets it, and the distance grows with the square root of height. Doubling the height does not double the distance; it multiplies it by about 1.41. That is why a mast or a hill adds so much range compared with standing on the beach.
To see something that rises above the surface, its own horizon counts too. A tall lighthouse can be seen from far away because two horizons add. The part of the object below the line of sight is hidden behind the curve, which is what the third mode measures.
Real conditions
Air bends light slightly downward, so the real horizon is often a bit farther than the geometric one, by roughly 7% on average, but it changes with temperature layers and can mirage or collapse. Waves, haze and terrain hide more. For navigation use proper charts and the hull speed calculator and tide rule of twelfths calculator for related boating figures, and the great circle distance calculator for distances between places.
The page treats the Earth as a sphere with the radius you choose. The tiny flattening of the planet changes the result by less than a third of a percent.
Assumptions and limits
- The surface is a smooth sphere and nothing blocks the view. Terrain, waves and haze are ignored.
- The default light path is straight. The coefficient 0.13 is an example, not a fixed property of the air.
- Distances are measured along the surface, which differs from the straight line of sight by less than one part in a million for these heights.
- Earth radius is a mean value of 6,371 km, adjustable from 6,300 to 6,400 km.
Frequently asked questions
How far away is the horizon for a person standing on the beach?
For an eye height of about 1.7 meters it is roughly 4.65 km, or 2.9 miles. This is why ships appear to rise from the sea only a few miles out, with their hulls hidden by the curve.
Does refraction make the horizon closer or farther?
Usually farther. Light bends along the curve of the Earth, as if the planet were a little flatter. The effect is a few percent, and it changes with the weather.
What does the hidden height mean?
It is how tall an object would need to be, at that distance, for its top to be just visible. Anything shorter than this, measured from the surface, is concealed behind the curve of the water or ground.
Can I use this to test whether the Earth is round?
The calculator assumes a sphere, so its results match what you would see on a spherical Earth with some bending of light. Comparing with measured sightings needs careful accounting of the air conditions on the day.