Horizon Distance Eye Height Calculator

Horizon distance from eye height

If you stand at height h (in metres) over a calm sea, the geometric distance to the horizon works out to d = √(2·R·h), with the Earth radius R ≈ 6371 km. Real air bends light downward (standard refraction), which acts as if the Earth radius were about 14% larger, so in practice people use d (km) ≈ 3.57 · √h (m). Stand on the beach with your eye at 1.7 m and the horizon sits roughly 4.7 km out. Climb a 30 m mast and it pushes to about 20 km. From Christ the Redeemer at 710 m you can see close to 95 km on a clear day, which is exactly why Niterói and even Petrópolis come into view.

Applications

Coastal navigation leans on it heavily, since the visible range of a lighthouse is the observer horizon plus the lamp horizon. The same math decides where to put lighthouses and signal towers, as places like Cabo Frio and Punta del Este do. It also matters for microwave and VHF radio links, for landscape photography from belvederes such as Alto da Boa Vista or Pedra Bonita, and for military lookout posts on ships and aircraft.

FAQ

Why include refraction? Air is denser close to the ground, so light rays bend downward and you end up seeing about 8% farther than the bare geometric figure suggests.

How do I find the range of a lighthouse? Add the two horizons together. An eye at 5 m reaches 8 km and a 50 m lamp reaches 25 km, so on a clear night the light shows up to roughly 33 km away.

Does temperature matter? It does. Strong thermal inversions create superior mirages that can more than double the apparent horizon, while very hot surfaces produce inferior mirages that pull it in.