How this instrument works
Your line of sight grazes a sphere, and that single fact fixes everything. Draw a right triangle: one leg runs from Earth's centre out to where your gaze just touches ground, length R. The hypotenuse runs from that same centre up through your eye, length R + h. Pythagoras hands over the remaining leg without ceremony — d² = (R + h)² − R² = 2Rh + h² — and that leg is your sight line. Since h is minuscule beside R, its square is nearly decorative: at eye level it contributes 2.89 m² against 21,661,400 m², roughly one part in seven and a half million.
Al-Biruni ran this triangle backwards around 1020 CE. From a hill of measured height near Nandana fort, in what is now Pakistan, he sighted a distant horizon, measured how far below level it sat, then solved for R — landing within a few percent of a modern value armed with a quadrant and trigonometry alone. Sailors had owned a qualitative version much longer: a departing hull vanishes before its mast, which is this same tangent line sweeping upward as range grows.
Two assumptions live inside d = √(2Rh + h²). It treats our planet as a smooth sphere of radius 6,371,000 m, an IUGG mean value; the genuine article is flattened by about one part in 298, and ground in front of you is seldom smooth or reliably at sea level. It also assumes light travels straight, which light declines to do — density gradients low in air bend rays gently downward, pushing a visible edge several percent beyond bare geometry. Only a clean sea horizon on a calm day really honours both conditions, which is why navigators trust this number more readily than hikers do.
- Measure to your eyes, not to your feet, and enter that figure under Eye height above the surface — cm, m, km and ft are all accepted.
- Read Distance to the horizon underneath, then pick its unit: km for land, miles for road maps, nautical miles for anything with a chart table.
- For a target of some height itself, run this sheet twice and add both answers; that sum is roughly when you and a lighthouse first see each other.
- Add about 8% by hand if you want a refracted, real-atmosphere edge rather than pure vacuum geometry.
Worked example — barefoot on a flat beach
Set Eye height above the surface to 1.7 m, roughly an adult's eye level standing on sand. Distance to the horizon returns d = √(2 × 6,371,000 × 1.7 + 1.7²) = √21,661,402.89 = 4654.1812 m. Call it 4.65 km, or 2.89 miles, or 2.51 nautical miles — a figure every navigation primer quotes for someone upright at sea level.
That is nearer than most people guess. A cargo ship five kilometres out already has its waterline tucked behind curvature. Notice also how little h² earned its place: discard that term and you get 4654.1809 m, an error of three tenths of one millimetre. Climbing pays far better than squinting — at 100 m your edge opens to 35.7 km, since doubling range demands four times as much height.
Questions
Is this how far away I can see things?
No — it is how far away a surface disappears. Anything tall enough sticks up past your horizon and stays visible well beyond it. Mount Teide on Tenerife, 3,715 m high, carries its own 218 km horizon circle; where two such circles overlap, two observers can see each other. Add both distances to get that combined sighting range, which is exactly how lighthouse geographic-range tables are built.
Does atmospheric refraction change my answer?
Yes, and always in one direction. Air thins with altitude, so rays curve gently toward ground and reach further than straight lines would, typically extending a horizon by 7–9% under ordinary conditions. Surveying and radio practice absorb this by using an effective radius of 7R ⁄ 6, which turns 3.57 √h into 3.86 √h with h in metres. Over hot sand or an ice sheet, temperature inversions can push that figure far higher, occasionally producing mirages of ships already below curvature.
Does this work on the Moon or Mars?
Yes — only R changes. Feed in a lunar mean radius of 1,737,400 m and eyes 1.7 m up reach barely 2.43 km; Mars, at 3,389,500 m, gives 3.39 km. Apollo crews remarked on how tight and close that lunar edge felt, and airless ground brings a bonus: with no atmosphere worth speaking of, no refraction correction applies, so bare geometry there is the whole truth. Mars retains just enough thin air to bend rays a little, though nothing like Earth manages.
Is d measured through air or along ground?
Through air, along a straight tangent. Arc distance hugging a surface is s = R·arccos(R ⁄ (R + h)), always slightly shorter than d. At human heights nobody could measure a difference — for h = 1.7 m both agree inside one millimetre. Only at orbital altitude does a real gap open: from 408 km up, tangent length reaches 2,316 km while ground-track radius is nearer 2,222 km.
How high must I climb to see 100 km?
Rearranged, h = d² ⁄ (2R), which gives 785 m for a 100 km geometric horizon — a modest mountain, or a tall broadcast mast. Square-root growth is unforgiving in one direction and generous in another: ten times further costs a hundred times more height, but your first metre off ground already buys 3.6 km.
Why do navigation tables print 1.17 √(height in feet)?
That constant delivers nautical miles directly from feet, and it quietly bakes refraction in. Pure geometry alone would use about 1.06. With eyes 1.7 m up — 5.58 ft — the table gives 2.76 nmi against a geometric 2.51 nmi. Bridge crews prefer the larger, refracted value because it matches what lookouts genuinely report on an average day.