From a height, how far away is the horizon? And how tall does something have to be to show above it? The curve of the Earth, with and without the bending of light in the air.
How to use it
Enter your eye height above the sea or the plain.
Enter the height of the object you are looking for, if any, and the distance to it.
Read the horizon distance, the greatest distance the object can be seen from, and how much of it the curve hides.
Choose a place to draw the horizon circle on the map.
5 practical uses
Work out how far a lighthouse, a mast or a ship can be seen from.
Check whether a distant island or mountain can be seen from a beach or a hill.
Plan the height of a camera, a radio mast or a lookout for the range it needs.
Explain the curve of the Earth with a real number for a class or a debate.
Estimate how far the horizon is from an aircraft or a balloon.
The maths behind it
From height h on a sphere of radius R, the tangent line to the surface has length d = √(2Rh + h²); for small h that is √(2Rh), about 3.57 km × √h in metres. Refraction is allowed for by using an effective radius of 7R/6, giving about 3.86 km × √h. An object of height h₂ is visible up to d(h₁) + d(h₂). At a distance x beyond the horizon the surface has dropped by √(R² + x²) − R.
Air bends light downward a little, so you see slightly farther than pure geometry allows. The usual allowance is an effective Earth radius 7/6 of the real one, which the tool applies in the "with refraction" figures.
How far is the horizon from the beach?
For eyes 1.7 m above the water, about 4.7 km (2.9 miles) with refraction.
Why can I see a mountain beyond the horizon?
Only the part above the horizon line is hidden. A tall object rises above it; the tool gives the greatest distance at which its top still shows.
Embed this tool on your website
Paste this code where the tool should appear. It shows the tool alone, without this page's text, and links back here. It is free to use; the map data credit stays on the map.
Change height to suit your page. To open the tool with a place already set, add the settings from this page's address bar (the part after #) to the end of the src. The embedded tool asks the visitor's browser for their location only when they press the button, and the allow attribute lets that work inside the frame.