Editing Horizon (section)

=== Approximation ===
[[File:horizon_distance_graphs.svg|thumb|250px|Graphs of distances to the true horizon on Earth for a given height ''h''. ''s'' is along the surface of the Earth, ''d'' is the straight line distance, and ''~d'' is the approximate straight line distance assuming ''h'' &lt;&lt; the radius of the Earth, 6371 km. In [http://upload.wikimedia.org/wikipedia/commons/4/44/Horizon_distance_graphs.svg the SVG image], hover over a graph to highlight it.]]
<!-- [[File:How far away is the horizon.png|thumb|right|300px]] -->
If the observer is close to the surface of the Earth, then ''h'' is a negligible fraction of ''R'' and can be disregarded the term {{nowrap|(2''R'' + ''h'')}}, and the formula becomes-

:<math>d = \sqrt{2Rh} \,.</math>

Using kilometres for ''d'' and ''R'', and metres for ''h'', and taking the radius of the Earth as 6371&nbsp;km, the distance to the horizon is

:<math>d \approx \sqrt{2\cdot6371\cdot{h/1000}} \approx 3.570\sqrt{h} \,</math>.

Using [[imperial units]], with ''d'' and ''R'' in [[statute mile]]s (as commonly used on land), and ''h'' in feet, the distance to the horizon is

:<math>d \approx \sqrt{2\cdot3963\cdot{h/5280}} \approx \sqrt{1.5h} \approx 1.22 \sqrt{h} </math>.

If ''d'' is in [[nautical mile]]s, and ''h'' in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:

:<math>d \approx \sqrt h </math>

These formulas may be used when ''h'' is much smaller than the radius of the Earth (6371&nbsp;km or 3959&nbsp;mi), including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both the metric and imperial formulas are precise to within 1% (see the next section for how to obtain greater precision).
If ''h'' is significant with respect to ''R'', as with most satellites, then the approximation is no longer valid, and the exact formula is required.