Table of Contents
Distance to the Horizon
The distance to the visible horizon is determined by the observer's height above the Earth's surface and the Earth's curvature. From sea level (eye height about 1.7 m), the horizon is approximately 4.7 km away. As height increases, the line of sight extends further before Earth's curvature blocks the view, making horizon distance proportional to the square root of height.
This calculation is important in navigation, aviation, telecommunications (line-of-sight radio), lighthouse design, and surveillance. Sailors use horizon distance to estimate when they'll spot land. Telecommunications engineers use it to determine maximum line-of-sight transmission distance between towers. Understanding horizon geometry is fundamental to mapping, geodesy, and even photography composition.
Formula
Where d is the distance to horizon, R is Earth's radius (6,371 km), and h is observer height. The approximation works when h is much smaller than R, which is true for all practical observer heights on Earth.
Horizon Distance by Height
| Height | Example | Horizon Distance |
|---|---|---|
| 1.7 m | Standing person | 4.7 km |
| 10 m | Ship deck | 11.3 km |
| 30 m | Lighthouse | 19.6 km |
| 100 m | Cliff top | 35.7 km |
| 10,000 m | Cruising aircraft | 357 km |
Atmospheric Refraction
Atmospheric refraction bends light rays around Earth's curvature, extending the visible horizon by about 8% beyond the geometric calculation. The standard refraction coefficient is approximately 0.13, giving an effective Earth radius of about 7,160 km for visibility calculations. In unusual atmospheric conditions (temperature inversions), extreme refraction can create mirages and dramatically extend or reduce the visible horizon.
FAQ
How far can I see on a clear day?
Visibility of distant objects depends on both horizon distance and atmospheric clarity. From a height of 2 meters, the geometric horizon is about 5 km. However, tall objects beyond the horizon can still be seen because their height extends above the curvature. A mountain 3000 m tall could theoretically be visible from nearly 200 km, though atmospheric haze usually limits visibility to 30-50 km.
Does the Earth's oblateness matter?
Earth is slightly oblate (equatorial radius 6,378 km vs polar 6,357 km). This 0.3% difference has negligible effect on horizon distance calculations for most purposes. However, for precision geodetic work and satellite observations, the WGS84 ellipsoid model is used rather than a simple sphere.
How is this used in telecommunications?
Line-of-sight radio communication requires that both antennas can "see" each other above the horizon. The maximum distance between two towers of heights h1 and h2 is approximately sqrt(2*R*h1) + sqrt(2*R*h2). This is why cellular towers are placed on high ground and why taller towers provide greater coverage areas.