Earth curvature calculator – distance to the horizon and hidden height

From a beach with your eyes 1.7 m above the water, the horizon of a calm sea is about 4.7 km away without refraction and 5 km with the usual bending of light in the air; from a 100 m cliff it is 36–38 km, and from the top of Mount Everest about 340 km. Anything farther away than the horizon starts to disappear from the bottom up: a 20 m ship 30 km off the coast, a lighthouse across a strait or the skyline of a city on the other side of a lake. This calculator answers the questions people ask about that: how far is the horizon, how far can I see from a given height, how much does the Earth curve over 10 km or 10 miles, how much of a far object is hidden, and how high must I stand to see its top. Type your eye height and a distance, or click where you stand and then click what you want to see, and the panel shows the horizon distance along the surface and in a straight line, the dip of the horizon below eye level, the drop of the surface, the hidden and visible part of the target, the eye height needed and the angle to its top. With "ground height" on, the heights of both points come from the same terrain model as the elevation finder, so a viewpoint on a hill counts. The horizon is drawn on the map as a ring, the table shows the curvature at 1 to 500 km or miles, and the result can be shared as a link or downloaded as GeoJSON or KML.

How to use it

1. Choose metres and kilometres or feet and miles, and type the height of your eyes above the ground.
2. Search your town, use your location or click the map to place yourself (A); the horizon ring appears.
3. Click what you want to see (B), or type its distance, and type its height, e.g. a ship's mast or a tower.
4. Read the hidden and visible part of the target, the eye height needed to see its top and the angle to it.
5. Change the refraction, open "Curvature by distance", copy the share link or download the horizon as GeoJSON or KML.

How accurate is it?

How the curvature is calculated

The Earth is treated as a sphere whose radius is the radius of curvature of the WGS84 ellipsoid at your position in the direction of the target, from about 6,335 km north–south at the equator to 6,400 km near the poles; without a target the mean over all directions is used, and without a place the mean Earth radius of 6,371 km. The horizon distance is R·arccos(R/(R+h)) along the surface, the hidden height of a target is the height of your line of sight above the surface where the target stands, and the drop is the same for an eye at the surface. The formulas are exact on the sphere and were checked against plain vector geometry to a micrometre; replacing the ellipsoid by its local sphere changes the results by far less than refraction.

Refraction bends the line of sight

Air is denser near the ground, so light bends slightly downwards and follows the curve of the Earth a little. Surveyors model this with a refraction coefficient k and an effective Earth radius R/(1 − k). The standard value for light, k = 0.13, makes the horizon about 7 % farther away and hides about 15 % less of a distant target; radio waves are usually modelled with k = 0.25 (the 4/3 Earth). Over cold water on a warm day k can be 0.5 or more, and over hot ground it can be negative, so the real hidden height can differ from the calculation by tens of percent. Mirages, looming and superior mirages bring objects into view that should be hidden; choose "None" for the pure geometry.

Ground heights and what is not included

With ground height on, the height of A and B comes from Terrain Tiles (SRTM and other surveys, typically within 5–10 m) and is added to the eye and target heights; sea and lake surfaces count as 0 m. The calculator then assumes a smooth Earth at sea level between the two points, which is right over water and flat land. Hills, forests and buildings in between are not checked: use the line of sight tool for that, which samples the terrain along the path. Waves, haze and the size of the target also decide whether you really see it.

Good to know

Drop is not the hidden height

The popular "8 inches per mile squared" gives the drop of the surface below a level line from a point at the surface: 66.7 ft at 10 miles. It is not how much of a distant object you cannot see, because your eyes are above the ground and the horizon is miles away. From 6 ft above the water, a target 10 miles away hides about 33 ft without refraction and 27 ft with it, half the drop.

Eye height, not standing height

Measure the height of your eyes, not the top of your head, above the water or the ground: about 1.6–1.7 m for a standing adult, plus the height of the pier, deck or cliff. A boat deck 3 m above the water with your eyes 1.7 m above it gives 4.7 m, and the horizon moves from 4.7 km to 7.7 km.

The horizon is not where the view ends

The horizon is where the sea surface disappears, not the limit of what you can see. A mountain or a tall tower beyond it can still be seen if its top rises above your line of sight: from 2 m above the sea a 100 m cliff is visible up to about 41 km away. The tool gives this as "farthest distance the top can be seen from".