page_title: Horizon
section_title: Derivation
hierarchical_section_title: Horizon / Distance to the horizon / Derivation
caption_attribution_description: English: Circle, chord and tangent to expalin OA.OB = OC^2
context_page_description: The horizon or skyline is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not.
  The true horizon is actually a theoretical line, which can only be observed when it lies on the sea surface. At many locations, this line is obscured by land, trees, buildings, mountains, etc., and the resulting intersection of earth and sky is called the visible horizon. When looking at a sea from a shore, the part of the sea closest to the horizon is called the offing.
  The true horizon is horizontal. It surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly spherical model of the Earth. Its center is below the observer and below sea level. Its distance from the observer varies from day to day due to atmospheric refraction, which is greatly affected by weather conditions. Also, the higher the observer's eyes are from sea level, the farther away the horizon is from the observer.
context_section_description: 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 secant-tangent theorem states that
  Make the following substitutions:
  d = OC = distance to the horizon
  D = AB = diameter of the Earth
  h = OB = height of the observer above sea level
  D+h = OA = diameter of the Earth plus height of the observer above sea level,
  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
  d = distance to the horizon
  h = height of the observer above sea level
  R = radius of the Earth
  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.
caption: Geometrical basis for calculating the distance to the horizon, secant tangent theorem
exif: {}