Distance To Horizon Calculator

What the distance to horizon calculator does

How far can you see before a planet's curve hides the rest? This calculator finds the distance to the horizon from your height above the surface, and works for any celestial body, with presets for the planets and moons. It can also solve the other way, finding the height or the body's radius from a horizon distance.

Below is what the horizon distance is, the equation behind it, why extra height matters less as you climb, and a worked example.

How to use it

1. Choose what to calculate: distance to the horizon, your height, or the body's radius.
2. Enter the values you know, picking a celestial body or entering a custom radius.
3. Press Calculate for the result, or Reset to clear it.

What the horizon distance is

The horizon is the line where the ground appears to meet the sky, and the distance to it is how far away that line lies. It exists because the surface of a planet curves: as you look outward, the ground gradually drops away below your line of sight, and beyond a certain distance it falls out of view entirely. That limit is the horizon. The taller you stand, the farther over the curve you can see, which is why a lookout climbs the mast and why lighthouses are built high.

For a person standing at the seashore, the horizon is only a few kilometres away, closer than most people expect. The distance depends on just two things: how high your eyes are above the surface, and how large the body you are standing on is. Because the calculation is pure geometry, it works equally well on Earth, the Moon, or any planet. This calculator computes the horizon distance from your height and the body's size.

The equation it uses

The horizon distance comes from a right triangle formed by the centre of the body, your eye, and the horizon point, solved with the Pythagorean theorem:

d = √((R + h)² − R²) = √(2Rh + h²)

Here d is the distance to the horizon, R is the radius of the body, and h is your height above the surface. The line of sight to the horizon just grazes the surface, meeting the radius there at a right angle, which is what makes the triangle work. Because your height is usually tiny compared with the planet's radius, the formula is very close to the simpler d ≈ √(2Rh), but the calculator uses the full expression for accuracy at any height.

Why height helps less and less

A striking feature of the horizon distance is that it grows with the square root of your height, not in direct proportion. This means doubling your height does not double how far you can see; it extends the horizon by only about 41 percent. Each extra metre of elevation adds less distance than the last, so the gains diminish as you climb.

The reason is the curvature itself. Low down, a small rise in height lets you peer a useful way over the gentle near curve. Higher up, you are looking down at an increasingly curved surface from a steeper angle, so each additional metre buys less new ground. This is why a modest hill already extends your view greatly compared with standing on the beach, but climbing a mountain ten times higher does not give you ten times the horizon, only a few times more. The calculator captures this square-root law exactly.

Horizons on other worlds

Because the horizon distance depends on the size of the body, it changes dramatically from world to world. On a small body, the surface curves away sharply, so the horizon is close; on a large one, the surface is gentler and the horizon is farther. From the same eye height, you would see much farther on a big planet than on a small moon.

On the Moon, whose radius is about a quarter of Earth's, the horizon at standing height is only about 2.5 kilometres away, noticeably closer than Earth's. On giant Jupiter, you could see more than three times as far as on Earth. On a tiny asteroid, the horizon might be just a stone's throw away, the surface curving down almost at your feet. The calculator's presets cover the Sun, the planets, and the Moon, letting you explore how the horizon shrinks or stretches depending on the world beneath you.

Units and precision

The calculator takes heights and distances in units from millimetres to nautical miles, and lets you select a celestial body or enter a custom radius, which suits everything from a person at the seashore to a satellite high above a planet. It uses the exact geometric formula, so the results hold even at large heights. The calculation assumes a smooth spherical body and ignores atmospheric refraction, which in practice lets you see slightly farther than pure geometry predicts.

A worked example

Suppose you stand on a beach on Earth with your eyes about 1.75 metres above the water, on a body of radius about 6,371 kilometres.

The horizon distance is d = √((R + h)² − R²) ≈ 4.7 kilometres, closer than many people guess. Climb to the top of the Cliffs of Dover, about 100 metres up, and the horizon stretches to about 35.7 kilometres. On the same beach but on the Moon, with its smaller radius, the horizon at eye level would be only about 2.5 kilometres, while on Jupiter it would reach about 15.6 kilometres.

Questions people ask

How do you calculate the distance to the horizon?

Use d = √((R + h)² − R²), where R is the body's radius and h is your height. For small heights this is very close to d ≈ √(2Rh).

How far is the horizon at the beach?

About 4.7 kilometres for a person standing at the shore on Earth, closer than most expect. Raising your viewpoint, as on a cliff or ship's mast, extends it considerably.

Why doesn't doubling my height double the horizon?

Because the horizon grows with the square root of height. Doubling your height extends the horizon by only about 41 percent, with each extra metre adding less than the last.

Is the horizon closer on the Moon?

Yes. The Moon is smaller than Earth, so its surface curves away faster, putting the horizon at about 2.5 kilometres at eye level, closer than Earth's 4.7 kilometres.

References

A quick note on where the geometry comes from. The horizon distance follows from the Pythagorean theorem applied to a right triangle at the tangent point, a standard result in astronomy and surveying, described clearly by San Diego State University's atmospheric optics resource. NASA provides the planetary radii used in the presets.

1. San Diego State University, Andrew Young, Distance to the Horizon. https://aty.sdsu.edu/explain/atmos_refr/horizon.html
2. NASA, Planetary Fact Sheet (radii of the planets and Moon). https://nssdc.gsfc.nasa.gov/planetary/factsheet/
3. OpenStax, University Physics Volume 1, geometry and the Pythagorean theorem. https://openstax.org/books/university-physics-volume-1/pages/2-1-scalars-and-vectors