Paritian

Science

Distance to the Horizon Calculator

How far the horizon is from any height, with and without atmospheric refraction, and how far away a tall object stays visible.

Results

Distance to the horizon 5.0271 km
The same in metres 5027.1 m
Without refraction 4.6542 km
Its own horizon distance 0.0000 km
How far apart you can still see it 5.0271 km
How far below level the horizon sits 0.0419 °
The same in nautical miles 2.7144 NM
How much of it the curve hides 15.081 m

What this tool does

From eye level the horizon is under five kilometres away, which surprises most people. Air bends light downwards a little, so the real horizon is further than pure geometry says — by about eight per cent under ordinary conditions, and more over cold water. Both numbers are given here, along with the trick that matters at sea: a tall object is visible from the sum of its horizon and yours.

Formula

d = √(2 R h + h²) · with refraction use R × 7/6 · range between two points = d(h) + d(H)

Variables

SymbolMeaningUnit
hHeight of your eyesm
hoHeight of what you are looking atm
dsHow far away it iskm
rkAtmospheric refraction—
rrRadius of the planetkm
DHDistance to the horizonkm
DMThe same in metresm
DGWithout refractionkm
DTIts own horizon distancekm
TDHow far apart you can still see itkm
DDHow far below level the horizon sits°
NMThe same in nautical milesNM
HDHow much of it the curve hidesm

Worked example

  • Height of your eyes1.7 m
  • Height of what you are looking at0 m
  • How far away it is20 km
  • Atmospheric refraction1.1666667
  • Radius of the planet6371 km
  • Distance to the horizon5.0271 km
  • The same in metres5027.1 m
  • Without refraction4.6542 km
  • Its own horizon distance0.0000 km
  • How far apart you can still see it5.0271 km
  • How far below level the horizon sits0.0419 °
  • The same in nautical miles2.7144 NM
  • How much of it the curve hides15.081 m

Limitations

  • The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
  • The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.