Distance to Horizon Calculator — Far Sight
The horizon is not a place but a calculation: the point where your line of sight grazes the curving Earth. Enter your height above the sea (or the plain), and this calculator returns the distance to that grazing point — geometric, and as corrected by the atmosphere’s bending of light.
Pythagoras on a Sphere
The geometry is a single right triangle: Earth's centre to your eye is R + h, centre to horizon point is R, and the sight line — tangent to the sphere — completes the triangle, giving d = √(2Rh + h²) by Pythagoras. For any human height the h² is a rounding term and the rule condenses to the mariner's favourite: distance in kilometres ≈ 3.57 × √(height in metres), with the atmosphere's standard refraction (light bending gently Earthward, as if the planet were larger — effective radius 7R/6) stretching it to about 3.86 × √h. The calculator prints both figures because both are used honestly: the geometric one is the truth of the sphere, the refracted one is the truth of the sea, where lights and landfalls habitually appear a few percent beyond the geometric limit — looming, the sailors called it, and built their tables on it.
Worked Example: Ten Metres Up
The defaults stand you on a small ship's deck or a low cliff, eye at 10 m: d = √(2 × 6,371,000 × 10) = 11.29 km geometric, about 12.19 km with standard refraction. Drop to a swimmer's or a kayaker's 2 m and the world shrinks to 5.05 km (5.45 refracted) — the reason the sea feels intimate from a boat and vast from a headland: the visible disc's area grows with h itself, so doubling your height doubles the ocean you command (the radius grows only as √h; the area, as h). Climb to a 100 m cliff and the horizon retreats to 35.7 km; from a cruising airliner at 10,000 m it sits 357 km away — pilots watch the sun set twice as slowly as the groundlings below them, and this square root is the entire explanation.
Meeting Horizons: How Lights Are Listed
Two observers see each other when their horizon circles overlap — the widget's note works the canonical case: a 30 m lighthouse throws its light's horizon about 21.1 km (refracted), a small boat's lookout adds 5.5, so the light can be picked up near 27 km — and nautical charts' "nominal range" figures are exactly such sums, capped by the lamp's luminous intensity in clear weather. The same addition prices radar horizons (radio waves refract slightly differently, the famous 4/3-Earth model), VHF radio range between two aerials, and the lookout's cry of "sail ho": every sighting between two heights is two horizon distances added, and the calculator's single-height figure is the atom of them all.
The Lookout's Arithmetic of War and Peace
Navies institutionalised this square root: the lookout's height in a ship's fighting top was chosen by it, radar pickets are stationed by its radio cousin, and the submarine's periscope depth is a negotiation between seeing (an eye a metre above the waves commands about 3.9 km of refracted horizon) and not being seen. In peace the same sum schedules ferries' visibility, lifeguards' tower heights and the marketing of every "sea-view" apartment: the premium floors are buying √h, whether the brochure says so or not. The formula's commercial value has always been this — it prices a view before the tower is built, the mast raised, or the lighthouse lit.
Radio Horizons and the 4/3 Earth
Radio engineers run this page with a fatter planet: VHF and UHF waves bend in the atmosphere slightly more than light does, conventionally modelled as an Earth of 4/3 the true radius — so a 100 m aerial's radio horizon is about 41 km where its optical one is 38.6 (refracted), and two such aerials link across 82 km of grazing path. Broadcast coverage maps, radar siting and microwave link budgets are all this correction at work, and their phrase for the calculator's output — the "radio horizon" — acknowledges the borrowing. The geometry is unchanged (Pythagoras outranks us all); only the radius is negotiated with the medium, which is physics' usual way of keeping a good formula by adjusting its constants.
The Beachgoer's Version
Stand at the waterline anywhere on Earth and your horizon is the calculator at h ≈ 2: five kilometres of ocean in every direction, a disc of about eighty square kilometres that feels like the whole sea and is, by the planet's measure, a postage stamp. The figure explains the sea's psychology — why ships appear to rise as they approach (their hulls climbing over your horizon's curve in the order the lookout's arithmetic predicts), why the coast "suddenly" appears on a crossing, and why a headland walk feels like a revelation every hundred metres of climb. The formula's square root is the shape of coastal wonder itself, quantified. It is also, not coincidentally, the shape of diminishing returns everywhere in engineering: the first metres of height are the cheapest horizon ever sold, and every metre after costs the same and buys less — a bargain whose terms this page states before the concrete is poured.
Related Tools
More from the Physics Calculators hub: the Index of Refraction Calculator — the bending that produces the refracted figure — and the Circular Motion Calculator, for the planet whose curvature this page prices.
Frequently Asked Questions
Can I see another country across the sea?
Run the sum: you need the far coast's elevation as well as your own. From Dover's white cliffs (about 100 m, horizon 38.6 km refracted) the French coast near Calais is low but the hills behind it rise past 100 m (another 38.6) — the sum, about 77 km, comfortably exceeds the 33 km strait, which is why France is a routine sight on clear days and was a military fact for centuries. From a beach at eye level (5.5 km) the same strait is an empty horizon — elevation is the entire difference between a famous view and a rumour, and this calculator is how you check a claimed sighting before repeating it.
Why does the horizon dip below eye level?
Because the tangent point is below your horizontal: the dip angle is arccos(R ÷ (R + h)) — about 0.10° from a 10 m deck, growing to roughly 1° at 1,000 m and over 3° for the airliner — a small but real correction that sextant navigation applies to every altitude sight (the "dip of the horizon" table in the nautical almanac is this formula, tabulated). Surveyors and snipers apply cousins of it. The calculator prices the distance; the dip is its angular sibling, and both come from the same right triangle the derivation drew.
What about mirages and abnormal refraction?
The 7/6 factor is a standard atmosphere; real air, layered by temperature, bends light more or less — over cold water, inversions can duct light far beyond the standard horizon (superior mirage, the Fata Morgana, ships floating on air), while over hot roads the bending reverses (the inferior mirage's phantom water). The calculator prints the standard figure because it is the honest planning number; the mariner's rule is that sightings beyond it announce abnormal air, and generations of polar explorers logged phantom lands that were exactly this — horizons misbehaving by physics, not by error.
How far can I see on the Moon or Mars?
Swap the radius: the same formula with R = 1,737 km (Moon) gives a 2 m eye a horizon of only 2.64 km — the lunar surface feels close and small because it is — while Mars (R = 3,390 km) gives 3.68 km, and both airless worlds deny you even those distances' clarity at grazing angles... no: without air there is no refraction bonus, but also no haze, so the geometric figure rules alone. The calculator standardises on Earth; the formula's hospitality to other radii is the reader's to enjoy with a pocket computation, and mission planners for rovers do exactly this to site their cameras.
Does the tide or my exact eye height matter?
At small heights, proportionally yes: the square root means a 2 m eye over a 5 m tide range has mis-stated its horizon by up to a quarter — charts therefore reckon heights from a stated tidal datum, and light ranges are quoted conservatively. At cliff and aircraft heights the tide vanishes into the rounding. The practical discipline this page teaches: at deck level, count your height honestly (eye, not deck; add the swell's better half if you must), because √h punishes sloppiness exactly where small craft live — in the first few metres, where each metre buys the most horizon.
How did this first prove the Earth is round?
The horizon's height-dependence was the Greeks' working evidence: Eratosthenes' circle knew ships vanish hull-first because the sea's surface curves away under the sight line — the geometry this page computes — and the distance at which a known-height light disappears measures the radius itself (al-Biruni, on a mountain in 1017, combined a measured peak height with the horizon's dip to get Earth's radius within a percent). The calculator thus replays one of science's founding measurements: give it an honest height and an honest disappearance distance, solve backwards, and the planet's size falls out of a right triangle.