How far can you see?
Because the Earth is curved, the surface drops away from your line of sight. The horizon is the point where your line of sight just grazes the surface. The higher your eyes, the farther it is.
The formulas
- Geometric distance: d = √(2Rh + h²), with R = 6,371 km, the mean radius of the Earth.
- With standard refraction: d ≈ 3.86 × √h km with h in meters, or about 1.32 × √h miles with h in feet.
Refraction bends light slightly along the curve of the Earth and adds about 8% to the geometric distance. The calculator shows both values.
Worked example
You stand on a beach with your eyes 6 ft (1.83 m) above the water:
- Geometric distance ≈ 3.00 miles.
- With refraction ≈ 3.24 miles (5.22 km).
A small boat beyond that distance starts to disappear hull first, which is one of the oldest visible proofs that the Earth is round.
Seeing distant objects
A tall object has its own horizon. You can see its top as long as the two horizon distances add up to more than the distance between you:
Maximum distance = horizon(your height) + horizon(object height)
From a 6 ft eye height, the top of a 160 ft lighthouse is visible up to about 3.2 + 16.7 ≈ 20 miles away with standard refraction. Lighthouse ranges printed on nautical charts use the same principle, usually with an observer eye height of 15 ft.
Horizon distance by height
| Eye height | Horizon with refraction |
|---|---|
| 5 ft (1.5 m) | 3.0 mi (4.8 km) |
| 6 ft (1.8 m) | 3.2 mi (5.2 km) |
| 100 ft (30 m) | 13.2 mi (21 km) |
| 1,000 ft (305 m) | 42 mi (67 km) |
| 10,000 ft (3,048 m) | 132 mi (213 km) |
| 35,000 ft (10,668 m), airliner | 247 mi (398 km) |
Why reality differs
- Haze and dust usually limit visibility well before the horizon.
- Temperature inversions over cold water bend light more strongly and can make distant coasts or ships appear, sometimes stretched or upside down (looming and Fata Morgana).
- Hot surfaces bend light upward and create inferior mirages that hide the bottom of objects.
Other planets
On the Moon, with a radius of 1,737 km, a 6 ft astronaut sees the horizon only about 1.5 miles away, which is why lunar photos show such a close horizon.