How far is the horizon?

Understanding the apparent distance of the horizon

The horizon is the apparent boundary between the Earth and the sky. Its distance depends on the Earth's curvature, the observer's height, and atmospheric refraction. This distance can be calculated using a simple geometric formula without considering atmospheric refraction.

Simple geometric distance to the horizon

The distance to the horizon, on an ideal, perfectly spherical Earth without atmosphere, is calculated by \(\,d = \sqrt{2Rh + h^2}\,\) where \(R\) is the Earth's radius (\(\approx 6371\) km) and \(h\) is the observer's height. For example, at 2 m height, the geometric horizon is about 5 km away.

Detailed calculation of the horizon distance

Consider an observer at a height \(h\) above the Earth's surface. Let \(R\) be the average radius of the Earth (\(R \approx 6371\) km). The line connecting the observer's eye to the point of tangency on the surface is perpendicular to the Earth's radius at that point. The triangle formed by the Earth's center, the point of tangency, and the observer is right-angled. Applying the Pythagorean theorem: \( (R + h)^2 = R^2 + d^2 \) where \(d\) is the straight-line distance between the observer and the horizon. The ground distance, following the curvature, is: \( s = R \cdot \theta \) with \(\theta = \arccos\left( \frac{R}{R+h} \right)\). Combining: \( s = R \cdot \arccos\left( \frac{R}{R+h} \right) \) and for \(h \ll R\), we obtain the approximation: \( s \approx \sqrt{2Rh + h^2} \)

Calculating the horizon distance with atmospheric correction

In reality, atmospheric refraction slightly increases this distance because light rays bend toward the ground. Under standard conditions, this effect is modeled by replacing \(R\) with \(R / (1 - k)\) where \(k \approx 0.13\).

Comparative table of horizon distances by altitude

N.B.:
The Earth's horizon never completely exists once you leave the surface: the higher you go, the more of the Earth's surface you can see. From the ISS (408 km), about 3.2% of the total surface is visible (≈9 million km²). In geostationary orbit (36,000 km), exactly half of the planet is visible. To observe 99% of the surface at a glance, you would need to reach an altitude of about 21,700 km. Beyond that, the horizon recedes almost to the opposite edge of the Earth, but it only disappears completely at infinite distance.