Horizon Distance Calculator
The horizon is not a fixed place: it is the tangent point where your line of sight grazes the curving Earth, and it moves outward as you climb. Enter your height above the surface in meters or feet and this calculator returns the geometric distance to that point in kilometers, miles and nautical miles, plus the dip angle of the horizon below true horizontal. It is a curvature calculation, not a map measurement: it tells you how far you can see from a height, not how far apart two places are.
Distance to the horizon
Geometric horizon on a smooth spherical Earth (R = 6,371 km), with no refraction. Real-world visibility usually runs about 7 to 8% farther because the atmosphere bends light downward.
Quick Answer
From eyes 2 meters above the water, the geometric horizon is 5.048 km away (3.14 mi, 2.73 nm), and it sits 0.0454 degrees below true horizontal. Distance grows with the square root of height, so quadrupling your height only doubles your view: at 100 m the horizon is 35.696 km away. The calculator above is pre-filled with the 2-meter case.
Key takeaways
- Four times the height gives twice the distance. Height is a weak lever, which is why a few extra meters on a headland beat a tall pair of boots on the beach.
- This tool returns the geometric horizon: no atmosphere, no weather. On a real day you usually see about 7 to 8% farther.
- The dip angle is the correction sextant navigators subtract from every altitude sight, and it stays tiny until you reach aircraft altitudes.
- Two viewpoints see each other at the sum of their two horizon distances, not at double one of them, unless both heights happen to be equal.
- Quick Answer
- How to use the horizon distance calculator
- The geometry: a tangent line, a radius, and one square root
- Worked example and reference heights
- Refraction: why you usually see a little farther than this
- Horizon distance is not map distance
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the horizon distance calculator
Enter one height, pick its unit, and click Calculate. Results stay hidden until you do.
- Observer height: measure from the surface the horizon sits on to your eyes or viewpoint. At the coast that is height above the water, on a ship it is deck or bridge height above the waterline, and in a plane it is altitude. A person standing at the waterline has eyes roughly 1.7 to 2 m up, which is why 2 m is the pre-filled example.
- Height unit: meters or feet. Feet convert at exactly 0.3048 m per foot before anything else happens, so 100 ft is treated as 30.48 m.
- Read the four outputs: the headline distance in kilometers, the same distance in miles and nautical miles, and the dip angle in degrees. Copy or share the full set with the buttons inside the results box.
- Edge cases, stated plainly: a height of 0 is accepted and returns 0.000 km. A blank field returns "Please enter the observer height." and a negative height returns "Height cannot be negative." Reset restores the 2-meter example.
The geometry: a tangent line, a radius, and one square root
The whole calculation is one right triangle. Your eye, the center of the Earth and the tangent point where your sight line touches the surface form a right-angled triangle, and solving it gives the distance.
Dip angle = arccos( R ÷ (R + h) )
Quick forms: d (km) ≈ 3.57 × √h (m) | d (mi) ≈ 1.22 × √h (ft) | d (nm) ≈ 1.06 × √h (ft)
Two honest details about the math. First, the exact tangent length is √(2Rh + h²): the tool drops the h² term, the standard textbook simplification that the Institute of Physics teaching resource on horizon distance also uses when it writes the approximation for classroom heights. The dropped term is worth 0.000 km at 100 m, 0.004 km at 1,000 m and 0.14 km (0.04%) at 10,000 m, so it only starts to matter at satellite altitudes this tool is not aimed at. Second, the dip angle is not approximated at all: arccos(R ÷ (R + h)) is the exact geometry, and it is the same quantity navigators look up in the Nautical Almanac dip table. The table values run slightly smaller than this tool's, because the almanac folds average refraction into its corrections: at a 2 m height of eye the table gives about 2.5 arcminutes of dip, against the tool's purely geometric 2.72 arcminutes (0.0454 degrees).
Worked example and reference heights
The pre-filled run is a person standing at the waterline with eyes 2 m above the sea. Every line below is a tool output, and the reference table after it was produced by running the same tool at each height.
| Output at 2 m | Value |
|---|---|
| Distance to the horizon | 5.048 km |
| Distance in miles | 3.14 mi |
| Distance in nautical miles | 2.73 nm |
| Dip angle below horizontal | 0.0454 degrees |
| Observer height | km | mi | nm | Dip |
|---|---|---|---|---|
| 1.7 m (average eye level) | 4.654 | 2.89 | 2.51 | 0.0419° |
| 2 m (worked example) | 5.048 | 3.14 | 2.73 | 0.0454° |
| 10 m (small cliff or ship deck) | 11.288 | 7.01 | 6.10 | 0.1015° |
| 30 m (headland or lighthouse gallery) | 19.551 | 12.15 | 10.56 | 0.1758° |
| 100 m (hill or tower) | 35.696 | 22.18 | 19.27 | 0.3210° |
| 1,000 m (mountain viewpoint) | 112.880 | 70.14 | 60.95 | 1.0151° |
| 10,000 m (airliner cruise) | 356.959 | 221.80 | 192.74 | 3.2081° |
The bars below use the same tool values. Read them as a square-root curve flattening out: the first 100 m buys you 35.696 km, the next 9,900 m buys about nine times more distance for ninety-nine times more height.
5.048 km
Geometric horizon at 2 m eye height, the standing-at-the-shore case.
3.3134°
Dip angle at 35,000 ft, entered in feet in the tool: the horizon as a pilot sees it.
×2
Distance gain when height is multiplied by 4, the square-root rule in one line.
6,371 km
Mean Earth radius built into every result on this page.
Refraction: why you usually see a little farther than this
On a real day the visible horizon sits roughly 7 to 8% beyond the geometric one, because air bends light gently downward and your sight line follows the bend. This tool deliberately leaves that out and says so; here is how to adjust.
Surveyors fold standard refraction into the constant: d (km) ≈ 3.86 × √h (m) instead of 3.57 × √h. At 2 m that lifts the answer from 5.048 km to about 5.46 km. The effect is not a fixed law of nature: Andrew Young's reference page on the distance to the horizon at San Diego State University documents sightings far beyond any standard correction when temperature layers duct light along the surface, and days when the horizon closes in instead. Treat the geometric figure as the dependable floor and the 3.86 rule as the typical-day estimate.
One constant causes endless confusion, so here it is untangled. Bowditch's American Practical Navigator gives the visible horizon in nautical miles as 1.17 × √(height of eye in feet). That 1.17 already includes average refraction. The geometric constant on the same units, the one consistent with this tool, is 1.06 × √h: at 100 ft the tool returns 10.64 nm, while the Bowditch visible-horizon rule gives 11.7 nm. Neither is wrong; they answer slightly different questions. A published critique of the 2002 Bowditch edition adds a caution worth knowing: its companion constant for meters, 2.07, is inconsistent with 1.17 and should read about 2.12.
Horizon distance is not map distance
This page measures how far a sight line reaches over a curved Earth from a given height. It does not measure the distance between two places, and using it for that job gives nonsense.
If you need the separation between two points on the ground, the distance calculator is the tool built for that question. For a straight line between coordinate pairs on a flat plane, the Euclidean distance calculator applies the coordinate formula directly. Pilots asking a different altitude question, how far an aircraft can glide from a height, want the glide distance calculator, which trades altitude for range through lift and drag rather than geometry alone. And if the dip angle is what you actually need, for a sextant sight or a survey setup, the distance and angle calculator covers the angle side of the triangle. Same family, four different questions; this page owns exactly one of them.
What competing calculators miss
The strongest rivals each do one thing better and several things worse. The honest scorecard:
- FIRGELLI's distance-to-horizon calculator: the deepest rival, with a refraction coefficient input, a reverse height-from-distance mode, a two-observer mode and a visualizer. Its geometric core is the same √(2Rh) used here. The cost is density: dip angle and the plain geometric answer sit inside an engineering package. This page keeps one job, puts dip in the main results, and prints a seven-row reference table you can check against the tool itself.
- SimpleMapLab's horizon tool: runs the exact √(2Rh + h²) form and adds a terrain viewshed. Good work, but its headline numbers bake standard refraction in, so the pure geometric answer is never isolated, and terrain is a different question from curvature. Here the approximation is disclosed with its size at four heights, so you can judge it instead of trusting it.
- CalculatorsHub's horizon page: presents 3.57 × √h as the refraction-adjusted "apparent" formula. That is backwards: 3.57 is the geometric constant, and the refraction constant is 3.86, as the surveying rule shows. Its sibling height-distance page also mixes building-height trigonometry into horizon content, two topics that share no formula.
- BraveCalculator's horizon page: meters in, kilometers out, nothing else: no feet, no miles, no nautical miles, no dip. Its worked example prints 35.682 km for 100 m; the correct geometric value, and the one this tool returns, is 35.696 km.
- The two-observer mistake, made almost everywhere: guides routinely tell you to double the one-way distance for the range between two antennas or two ships. That only works when both heights are equal. The correct range is d(h1) + d(h2): run this tool once for each height and add.
Frequently asked questions
Horizon distance in kilometers is about 3.57 times the square root of your height in meters. It is this tool's formula with the arithmetic pre-done: the square root of twice Earth's radius of 6,371 km produces the 3.57 constant. At 2 m the rule gives 5.05 km, matching the tool's 5.048 km.
Yes. Air bends light downward, so the visible horizon usually sits about 7 to 8% beyond the geometric one this tool returns. Surveyors use 3.86 times the square root of height in meters to include standard refraction, which lifts the 2 m answer from 5.048 km to about 5.46 km. Unusual temperature layers can push the visible horizon much farther, or pull it closer.
It is the angle the visible horizon sits below true horizontal from your viewpoint. It is tiny at eye level, 0.0454 degrees at 2 m, and grows with height: at 35,000 ft the tool returns 3.3134 degrees, which is why the real horizon drops visibly below an airliner's artificial horizon line. Sextant navigators subtract dip from every altitude sight.
As a first approximation, yes. The range between two antennas or two ships is the sum of their separate horizon distances: run the tool for each height and add the results. Doubling one distance is only correct when both heights are equal. VHF radio waves also bend slightly with the atmosphere, so working radio range usually runs a little beyond the geometric sum.
Mariners and aviators work in nautical miles, where 1 nm equals 1.852 km. The geometric rule on those units is about 1.06 times the square root of height in feet. The better known 1.17 rule from Bowditch is a visible horizon rule that already includes average refraction, so it returns a larger figure than this geometric tool.
Enter 35000 with the unit set to feet and the tool returns 368.689 km (229.09 mi, 199.08 nm), with a dip angle of 3.3134 degrees. That is the geometric figure; on a typical day refraction stretches the visible horizon a few percent farther still.
Very accurate at everyday heights. The tool uses d = √(2Rh), which drops the small h² term from the exact tangent length √(2Rh + h²). The difference is 0.000 km at 100 m, 0.004 km at 1,000 m and 0.14 km at 10,000 m, about 0.04%, so it only matters at altitudes far above airliner cruise.
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References
- Young, Andrew T., San Diego State University. Distance to the Horizon. Accessed October 22, 2026.
- Institute of Physics, IOP Spark. Broadening Your Horizons: Distance to the Horizon. Accessed October 22, 2026.
- Stuart, Robin G. Bowditch 2002 Errors and (In)actions. NavList. Accessed October 22, 2026.
- Nautical Almanac Office, table reproduction. Correction Table for Dip of the Horizon. Accessed October 22, 2026.
- Murphy, Thomas, University of California San Diego. Physics 281 Lecture Notes: Atmospheric Refraction. Accessed October 22, 2026.
- Dutch, Steven, University of Wisconsin Green Bay. Earth Data: Dimensions. Accessed October 22, 2026.