Dip of the Horizon: How Far Can You See?
📐 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 δ ≈ 0.032 × √(h in metres) degrees (geometric, no refraction). Distance to the horizon d ≈ 3.57 × √(h in metres) km. Worked: at h = 10 m (a tall building) δ ≈ 0.10° and d ≈ 11 km; at h = 100 m (a hilltop) δ ≈ 0.32° (about the Moon's diameter) and d ≈ 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.