Distance to Horizon Calculator
Estimate the straight-line distance from an observer to the geometric horizon on Earth, another Solar System body, or a sphere with a custom radius.
Inputs
Required. Enter a non-negative decimal using a dot as the decimal separator.
Choose a preset radius, or select a custom radius for another spherical object.
Required only for a custom body. Radius must be greater than zero.
Changes presentation only; the geometric model is calculated in metres.
Distance to horizon
4.722 km
For a 1.75 m observer on Earth, the unobstructed geometric horizon is about 4.722 km away along the straight tangent line.
d = √(h × (2r + h))Calculation details
| Quantity | Meaning | Value |
|---|---|---|
| Observer height, h | Height above the average spherical surface | 1.750 m |
| Body radius, r | Radius used by the spherical model | 6,371,008 m |
| Observer radius, r + h | Distance from the sphere center to the observer | 6,371,009.750 m |
| Distance to horizon, d | Straight tangent-line distance to the horizon point | 4,722.132 m |
How to use the Distance to Horizon Calculator
What this calculator does
This calculator estimates the geometric line-of-sight distance from an observer to the horizon on a spherical body. It uses the observer's height above the average surface and the body's radius to form a right triangle whose tangent point is the horizon. The result is a straight-line distance from the observer to that tangent point. It does not model mountains, buildings, local terrain, waves, atmospheric refraction, or the detailed ellipsoidal shape of a real planet, so it is best treated as a clean geometric estimate rather than a guarantee of what will actually be visible.
When to use it
Use the calculator to estimate how far an unobstructed sea horizon lies from eye level, compare the effect of standing on a hill or tower, explore how horizon distance changes on the Moon or another planet, or test the same geometry for a large custom sphere. The planetary presets use representative mean radii. NASA's Solar System size reference provides a useful context for the planetary radii used in this kind of comparison.
How to calculate
- The calculator opens with a ready-to-use demonstration: Height above average surface level = 1.75 m, Celestial body = Earth, and Distance to horizon unit = km. The displayed result and the example Excel workbook are immediately available.
- Replace Height above average surface level with your observer height. Use the adjacent Height unit control to switch among metres, feet, kilometres, and miles; the typed value is converted when you change units.
- Choose a Celestial body. If you choose Enter a custom radius, the Custom radius and Custom radius unit controls appear. Enter a positive radius for the sphere you want to model.
- Choose the Distance to horizon unit. This changes only the displayed unit, not the underlying geometry. Read the primary Distance to horizon result and the Calculation details table, then use Download Excel if you want the current inputs and outputs in a workbook.
- Reset clears the demonstration height and any custom radius, returns unit and body controls to their neutral defaults, removes calculated content, and disables Excel export until a complete valid state is entered again.
Input guide
Height above average surface level is a required non-negative decimal. It represents the observer's eye, camera, antenna, or platform height above the body's average surface. A realistic example is 1.75 with the Height unit set to m. Higher values increase the horizon distance. Enter a plain decimal with a dot; commas, percent signs, scientific notation, and mixed-unit text are rejected rather than silently reinterpreted. A zero height is allowed mathematically and produces a zero horizon distance.
Height unit is required and may be m, ft, km, or mi. Changing it converts the current valid height, so 1.75 m becomes approximately 5.741 ft. Do not manually convert the number and then also switch the unit, because that would convert twice.
Celestial body is required. Earth is the startup choice; Moon, Mercury, Venus, Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto are available as presets. Selecting a larger radius generally pushes the horizon farther away for the same observer height. The presets are spherical approximations, not detailed geodetic models.
Custom radius is required only when Enter a custom radius is selected. It must be a positive finite decimal, for example 1000 km. Custom radius unit may be m, km, ft, or mi and converts a valid custom-radius value when changed. A radius of zero or a negative radius is invalid because the tangent geometry requires a real sphere.
Distance to horizon unit is a required presentation control with km, m, mi, and ft options. It does not change the canonical result; it only converts the same distance for display and export labels. The NIST overview of SI units explains why the calculator uses metres as its internal length unit.
Output guide
The header pills summarize Body, Radius, Height, and Horizon so you can confirm the current state at a glance. The primary Distance to horizon result is the straight tangent-line distance, formatted in the selected output unit. A zero result is valid only for zero observer height. For fixed radius, larger heights produce larger distances; for fixed height, larger bodies also produce larger tangent distances.
The Calculation details table exposes four canonical quantities: Observer height, h; Body radius, r; Observer radius, r + h; and Distance to horizon, d. The table is shown in metres to make the dimensional consistency of the formula explicit. These are calculated identities, not empirical measurements. The Quantity column names each variable, Meaning states its geometric role, and Value reports the canonical metre value. The formula line, d = √(h × (2r + h)), is algebraically equivalent to d = √((r + h)² – r²) but avoids subtracting two nearly equal large squares in floating-point arithmetic.
Worked example
With the startup values, Earth uses a radius of 6,371,008 m and the observer height is 1.75 m. Substituting into the right-triangle identity gives d = √(1.75 × (2 × 6,371,008 + 1.75)) = 4,722.132 m, which displays as 4.722 km. The same value appears in the first-open result and workbook. The geometric step comes directly from the Pythagorean theorem; OpenStax's right-triangle explanation of the Pythagorean theorem is a useful refresher.
How the horizon formula works
Imagine a line from the center of a spherical body to the observer and another line from the center to the point where the observer's line of sight just touches the surface. A radius drawn to a tangent point is perpendicular to the tangent, so those three points create a right triangle. The hypotenuse is r + h, one leg is r, and the other leg is the horizon distance d. The Pythagorean relationship therefore gives d² + r² = (r + h)². Expanding and simplifying yields d² = 2rh + h² and then d = √(h(2r + h)).
For ordinary viewing heights on Earth, the geometric model is highly sensitive to height because distance grows approximately with the square root of height. Doubling your eye height does not double the horizon distance; it increases it by roughly a factor of √2 when the height is tiny relative to Earth's radius.
Limits and interpretation
The result is a geometric horizon for a smooth sphere. Real visibility can be shorter because terrain, buildings, vegetation, sea state, haze, and other obstructions block the line of sight. In some atmospheric conditions, refraction bends light and shifts the apparent horizon, so observed distance can differ from the purely geometric value. Planetary bodies also are not perfect spheres; Earth is better represented by an oblate ellipsoid, and local elevation is not identical to height above an idealized average surface. These effects matter when survey-grade, geodetic, optical, or radio-propagation precision is required.
For everyday comparisons, however, the spherical model is an excellent way to understand the dominant geometry. NASA's planetary-size data show why the same eye height gives noticeably different horizons on different worlds: radius is part of the right-triangle relationship. Keep units consistent, use a realistic height relative to the selected body, and interpret the output as a model-based geometric distance rather than a visibility forecast.