How can altitudes be used to calculate the circumference of the earth?
See explanation for a nice approximation, using altitude H km and visible horizon ( nautical ) distance D km :
Let B be the beacon from the top of a Light House (LH), at a
height H meters, from sea level.
Let D be the nautical distance traveled by a ship S,
sailing away from the LH, and
center C of [the Earth](https://socratic.org/astronomy/our-solar-
When the beacon B just disappears beneath the horizon, from
the sea-level Telescope of the ship, then
BS touches the Earth at S, and so,
D = R
As D/R is small,
Now, H/R is small, and so, the RHS
For sample data H = 100 meters = 0.1 km, D = 11.3 km.
circumference = 40, 116 km, nearly.
Here, the assumed visible-horizon ( nautical ) distance D = 11.3 km,
against the altitude 100 meters,
For this formula and R = 6371 km, the visible-horizon (nautical )
distance, for the Statue
of Liberty of height H = 41 meters is
D = 22.9 km, nearly