Dip of the Horizon: How Far Can You See?
π Measuring the Earth
Question
How far below your horizontal does the horizon sit, and how far can you see from a height?
Stand on a sphere and the visible horizon sits below your horizontal line of sight. That angle - the dip - grows with your height. It is a direct, measurable consequence of the Earth's curvature, and it is how Al-Biruni measured the Earth's radius in 1024 (by measuring the dip from the top of a mountain). It is also in every navigation manual: the higher you are, the further you can see.
The math
Dip Ξ΄ β 0.032 Γ β(h in metres) degrees (geometric, no refraction). Distance to the horizon d β 3.57 Γ β(h in metres) km. Worked: at h = 10 m (a tall building) Ξ΄ β 0.10Β° and d β 11 km; at h = 100 m (a hilltop) Ξ΄ β 0.32Β° (about the Moon's diameter) and d β 36 km.
Why it matters
This is the formula behind the $10 horizon-dip experiment, and it is the same measurement Al-Biruni used to find the Earth's radius a thousand years ago. The dip is small at human heights (which is why the ground looks flat), but it is exactly what a sphere predicts, and it is measurable with a sub-$10 clinometer.