How it works
- Treat Earth as a sphere with a radius of 6,371 km (3,959 miles). The line of sight from height h just touches the surface at a distance of d = √(h × (2R + h)), almost exactly √(2Rh), since h is tiny next to Earth.
- Light bends slightly toward the ground because the air is denser lower down. Surveyors model this with a refraction coefficient k ≈ 0.13, which stretches the radius to R ÷ (1 − k). That adds about 7% to the distance.
- The result matches the horizon formula in the NGA's The American Practical Navigator (Bowditch), 2.07√h nautical miles with h in meters, to within 0.2%.
- Mutual visibility limit = your horizon distance + the other object's horizon distance, assuming no higher ground in between.
- Hidden height = (distance − your horizon distance)² ÷ (2R′). That much of the far object sits below the horizon.
Horizon distance by height
| Height | With refraction | Without |
|---|---|---|
| 5.5 ft | 3.1 mi (5.0 km) | 2.9 mi |
| 10 ft | 4.2 mi (6.7 km) | 3.9 mi |
| 30 ft | 7.2 mi (11.6 km) | 6.7 mi |
| 100 ft | 13.1 mi (21.1 km) | 12.2 mi |
| 300 ft | 22.7 mi (36.6 km) | 21.2 mi |
| 1,000 ft | 41.5 mi (66.8 km) | 38.7 mi |
| 1,250 ft | 46.4 mi (74.7 km) | 43.3 mi |
| 5,000 ft | 92.8 mi (149 km) | 86.6 mi |
| 14,505 ft | 158 mi (255 km) | 148 mi |
| 35,000 ft | 246 mi (395 km) | 229 mi |
Heights: Empire State Building 86th floor 1,050 ft, 102nd floor 1,250 ft, tip 1,454 ft (ESB); Willis Tower Skydeck 1,353 ft (operator); Denali 20,310 ft (USGS); Mount Whitney 14,505 ft and Mount Rainier 14,410 ft (National Park Service); airliners typically cruise at 30,000-40,000 ft.
Why you usually see less
These are theoretical limits for perfectly clear air. Dust and water vapor often cap visibility at a few dozen miles, and hills or buildings block the line of sight. On days with a temperature inversion, light bends more and you can see farther than usual (mirages). Refraction changes with the time of day and the weather, so read the result as a range between the two values.
Frequently asked questions
How far away is the horizon when you stand at the beach?
With your eyes about 5.5 feet above the water, the horizon is about 3.1 miles (5 km) away. Light bends slightly in the air, so you see a little farther than pure geometry says (2.9 miles). Sit on the sand with your eyes about 3.3 feet up and it shrinks to roughly 2.4 miles.
How far can you see from the Empire State Building?
From the 86th-floor deck at 1,050 feet, the horizon is about 43 miles away over flat ground or water; from the 102nd floor at 1,250 feet, about 46 miles. Haze usually limits the real view to less than that.
How far can you see from an airplane?
At a cruising altitude of about 35,000 feet, the horizon is roughly 246 miles away. Airliners typically cruise between 30,000 and 40,000 feet.
How far apart can two mountains see each other?
Add the horizon distance from each height. Two 10,000-foot peaks can see each other from up to about 263 miles apart (131 + 131), as long as no higher ground sits between them.
Why does the bottom of a distant tower disappear first?
Earth curves away, so the base drops behind the horizon. From eye height 5.5 feet, a 300-foot tower 20 miles away loses about its bottom 166 feet; only the top 134 feet stay visible. Enter a distance and this calculator shows the hidden height.