Distance to Horizon Calculator
Enter your height above the surface to find the geometric distance to the horizon. Switch between metric and imperial units, pick a celestial body (Earth, Moon, Mars and more), or enter a custom radius. Toggle atmospheric refraction to see the real-world correction, and use reverse-solve mode to find the height you need to spot a distant object.
Formula
Worked example
An adult standing 1.75 m above sea level on Earth (r = 6,371 km): d = sqrt((6371000 + 1.75)^2 - 6371000^2) = sqrt(2 * 6371000 * 1.75 + 1.75^2) = approx 4,718 m = 4.72 km. With the 7/6 refraction correction the effective radius becomes 7,433 km and the horizon extends to about 5.10 km.
How the horizon distance formula works
The horizon is the point where a straight line of sight from your eyes just grazes the surface of a sphere. Connecting the center of the sphere (C) to you (O) gives a line of length r + h, where r is the sphere radius and h is your height. The line from C to the horizon point (H) has length r, and the angle at H is exactly 90 degrees because the line of sight is tangent to the sphere. Pythagoras then gives d = sqrt((r + h)^2 - r^2). For small heights relative to the radius, this simplifies closely to d = sqrt(2rh), which is the familiar approximation. This calculator uses the exact formula so it remains accurate even for aircraft and satellites.
Atmospheric refraction and the 7/6 rule
Light traveling through air bends slightly toward the surface because the atmosphere becomes denser at lower altitudes. The net effect is that you can see slightly farther than pure geometry predicts: the beam curves enough to follow the curvature of Earth a little. A standard approximation replaces the geometric radius r with an effective radius of (7/6) r, increasing it by about 16.7%. This adds roughly 7% to the horizon distance at typical observation heights. Meteorological conditions change the exact factor, so the 7/6 value is a long-run average for standard atmospheric conditions. Toggle refraction on or off in this calculator to compare both results.
Horizon distances on other worlds
Because horizon distance scales with the square root of the planetary radius, smaller bodies have surprisingly close horizons. On the Moon (r = 1,737 km), an astronaut standing 1.75 m tall would see the horizon at only about 2.4 km, compared to 4.7 km on Earth. On Jupiter (mean radius 69,911 km), the same height yields about 15.6 km. This matters in practice: astronauts on the lunar surface reported that craters and hills appeared much closer than they expected, partly because the close horizon creates a foreshortened perspective with no atmospheric haze to provide depth cues.
Practical applications
Sailors have used the concept since antiquity to spot tall ships over the horizon before the hull becomes visible, a technique called "hull-down." Coastal lookouts, lighthouse designers and radar engineers size towers to ensure coverage over a required sea area. Aviation uses a similar calculation to determine the ground footprint visible from a given altitude and to plan line-of-sight radio links. Search-and-rescue operations use the reverse-solve direction: given a search area radius, how high must a helicopter fly to scan the full zone without gaps?
Typical horizon distances on Earth
| Observer / scenario | Height | Horizon distance |
|---|---|---|
| Child standing (eyes ~1.2 m) | 1.2 m (3.9 ft) | 3.9 km (2.4 mi) |
| Adult standing (eyes ~1.75 m) | 1.75 m (5.7 ft) | 4.7 km (2.9 mi) |
| Person on a 10 m cliff | 10 m (33 ft) | 11.3 km (7.0 mi) |
| Tall building (50 m floor) | 50 m (164 ft) | 25.2 km (15.7 mi) |
| Hilltop (200 m) | 200 m (656 ft) | 50.5 km (31.4 mi) |
| Aircraft cruise (~10 km) | 10,000 m (32,800 ft) | 357 km (222 mi) |
| Low Earth orbit (~400 km) | 400,000 m (1,312,336 ft) | 2,293 km (1,424 mi) |
Approximate horizon distances for common observation heights on Earth, using the standard geometric formula without atmospheric refraction.
Frequently asked questions
How far is the horizon from eye level?
For an average adult with eyes about 1.75 m (5 ft 9 in) above sea level, the geometric horizon on Earth is about 4.7 km (2.9 miles). With standard atmospheric refraction the distance extends to roughly 5.1 km (3.2 miles). The exact value depends on eye height and whether refraction is applied.
Why does the formula use sqrt((r+h)^2 - r^2) instead of sqrt(2rh)?
Both come from Pythagoras: d^2 = (r+h)^2 - r^2 = r^2 + 2rh + h^2 - r^2 = 2rh + h^2. The simpler form sqrt(2rh) drops the h^2 term, which is negligible for small heights (people, hills) but grows important for aircraft or satellites. This calculator keeps the full exact formula so it stays accurate at any height.
What is atmospheric refraction and does it matter?
Atmospheric refraction is the bending of light in the atmosphere, which curves the beam slightly downward. For typical weather conditions the effect adds about 7% to the horizon distance. It varies with temperature, pressure and humidity, so the 7/6 effective-radius rule is an average rather than an exact value. For most practical purposes the difference between the refracted and geometric results is small, but it becomes noticeable for long-distance sightings and radio line-of-sight calculations.
How do I find out how high I need to climb to see something?
Switch the calculation mode to "Distance -> Height (reverse solve)". Enter the distance to the target object, and the calculator returns the observer height you need. The reverse formula is h = sqrt(r^2 + d^2) - r, which is just Pythagoras rearranged. Remember that both the observer and the target have heights above the surface, so for a distant mountain peak you only need to account for the height difference, not the full elevation.
Does this work for the Moon or other planets?
Yes. Select a body from the dropdown (Earth, Moon, Mars, Jupiter, Saturn, Mercury, Venus) or choose Custom and enter any radius. The Moon has a radius of about 1,737 km, so the horizon is much closer than on Earth for the same observer height. Atmospheric refraction only applies to bodies with significant atmospheres, so on the Moon (essentially no atmosphere) you should leave refraction off.
Why does a ship disappear hull-first over the horizon?
As a ship sails away, the lower parts of it (the hull) disappear below the horizon before the taller parts (the masts or superstructure). This is direct visual evidence that Earth is spherical. The hull-down distance depends on the height of the observer and the height of the ship, and can be calculated by adding the individual horizon distances for each: d_total = sqrt(2 * r * h_observer) + sqrt(2 * r * h_ship).