Horizon
Derivation

If the Earth is assumed to be a featureless sphere (rather than an oblate spheroid) with no atmospheric refraction, then the distance to the horizon can easily be calculated.

The tangent-secant theorem states that

Make the following substitutions:

with d, D, and h all measured in the same units. The formula now becomes

or

where R is the radius of the Earth.

The same equation can also be derived using the Pythagorean theorem.At the horizon, the line of sight is a tangent to the Earth and is also perpendicular to Earth's radius. This sets up a right triangle, with the sum of the radius and the height as the hypotenuse. With

referring to the second figure at the right leads to the following:

The exact formula above can be expanded as:

where R is the radius of the Earth (R and h must be in the same units). For example,if a satellite is at a height of 2000 km, the distance to the horizon is 5,430 kilometres (3,370 mi);neglecting the second term in parentheses would give a distance of 5,048 kilometres (3,137 mi), a 7% error.
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