Standing on a beach, the line where the sea meets the sky feels endless, but it is only about 4.7 km away. A single square root tells you exactly how far you can see from a beach, a rooftop or a plane window, and why climbing four times higher only doubles your view.
Why the horizon has a fixed distance
The Earth is curved, so your line of sight eventually skims the surface and carries on into the sky. That point is your horizon, and it moves further away as your eyes go higher.
Geometry does the rest. Your line of sight, the Earth’s radius and the line from the Earth’s centre to your eyes form a right triangle, so Pythagoras’ theorem gives the distance directly. The exact version is d = √(2Rh + h²), where R is the Earth’s radius (about 6,371 km) and h is your height.
The simple formula you can do in your head
For any height you will reach as a traveller, h is tiny compared with the Earth’s radius, so the h² part can be dropped. What is left simplifies to:
Distance to the horizon (km) ≈ 3.57 × the square root of your eye height (in metres).
In plain words: take the square root of your eye height, then multiply by 3.57. In feet and miles, use 1.22 × √(height in feet). Both stay within about 1% of the exact formula, even from a mountaintop or an aircraft.
Real examples, from the beach to 35,000 feet
Here is what the formula gives, with no atmospheric effects:
- Standing on the shore, eyes at 1.70 m: √1.70 ≈ 1.30, × 3.57 ≈ 4.7 km
- A hill or tower 30 m above the sea: √30 ≈ 5.48, × 3.57 ≈ 19.6 km
- A viewpoint 100 m up: √100 = 10, × 3.57 ≈ 36 km
- The roof of the Burj Khalifa, about 834 m above sea level: ≈ 103 km
- The top of Mount Everest, 8,848 m: ≈ 336 km
- A plane cruising at 35,000 feet: ≈ 369 km
Notice the pattern. Going from 1.70 m to 100 m makes you almost 60 times higher, yet you only see about 8 times further. Because the distance follows a square root, four times the height gives you only twice the view.
Seeing things beyond your horizon
Tall objects stick up above your horizon. The rule is simple: add your horizon distance to the object’s own horizon distance.
Take a 100 m tower on the coast. From the beach your horizon is 4.65 km, and the tower’s is 35.7 km. You can therefore spot its top from up to about 40 km away, even though its base disappeared long before.
The same logic works in reverse to estimate distance. Say your eyes are 2 m above flat ground and you look at a 30 storey building, each floor 3.5 m high. Through binoculars you can count only 10 floors, so 70 m of it is hidden by the Earth’s curve. The distance is about 3.57 × (√2 + √70), which comes to roughly 35 km.
You can also work out how much of a ship is hidden. With your eyes 10 m above the sea, your horizon is 11.3 km away. A ship 20 km out is 8.7 km beyond it, and (8.7 ÷ 3.57)² ≈ 6, so everything below about 6 m on that ship sits below the waterline you can see.
The air lets you see a little further
The atmosphere bends light slightly downward, around the curve of the Earth. In standard conditions that pushes the visible horizon about 8% further, which changes the factor from 3.57 to about 3.86. On the beach, that turns 4.7 km into roughly 5 km.
In unusual weather this breaks down: over cold water the effect gets much stronger, while over hot desert ground light bends upward and creates mirages.
Common mistakes to avoid
- Assuming distance grows in a straight line with height. If 1.70 m gives 4.7 km, 100 m does not give 270 km, it gives 36 km.
- Mixing units. The 3.57 factor needs metres in and gives kilometres out. Feet need the 1.22 factor and give miles.
- Using it for satellites. At 2,000 km up, dropping h² gives 5,048 km instead of 5,430 km, a 7% error. Use the exact formula there.
- Forgetting obstacles. Hills, trees and buildings usually block your view long before the true horizon.
Try it with your own numbers
All you need is one square root. Plug your eye height into our square root calculator, multiply the result by 3.57, and you have your horizon in kilometres. For the distance between two places on the map rather than what you can see, read our guide to great-circle distance.
