Geometric distance to the horizon
For an observer above a spherical surface, the line of sight to the geometric horizon is tangent to Earth. The radius, observer height, and tangent distance form a right triangle.
This result is an ideal straight-line distance to the tangent point. Real visibility depends on atmospheric refraction, terrain, Earth’s ellipsoidal shape, target height, weather, and visual contrast.
How to calculate horizon distance
- Enter observer height: Use eye or sensor height above the local surface in meters.
- Calculate: Read the ideal straight-line distance to the tangent horizon.
- Apply real-world corrections: Account separately for terrain, refraction, target height, and local Earth geometry when needed.
Formula and variables
The calculator uses height in meters, a mean Earth radius of 6,371,000 m, and reports straight-line distance in kilometers.
d = √((R + h)² − R²) = √(2Rh + h²)- d — Horizon distance
- Straight-line distance from observer to tangent point (km)
- R — Earth radius
- Spherical mean-radius approximation (m)
- h — Observer height
- Height above the modeled surface (m)
Eye height of two meters
An observer’s eye is 2 m above a level spherical surface.
- Height
- 2 m
- Earth radius
- 6,371 km
- d = √(2 × 6,371,000 × 2 + 2²)
- d ≈ 5,048 m
Result: The geometric horizon is approximately 5.05 km away.
Refraction often increases apparent range, while terrain or obstructions can reduce it.
Understanding your results
Know which distance is reported
The calculator reports the straight tangent line, not surface arc distance.
- At ordinary observer heights, straight and arc distances are very similar.
- A target above the surface has its own horizon contribution.
- Optical and radio horizons can differ because refraction depends on conditions and frequency.
Assumptions
- Earth is a sphere of radius 6,371 km.
- Observer height is measured radially above the surface.
- The line of sight is unobstructed and refraction-free.
Limitations
- Does not model atmospheric refraction, terrain, ellipsoidal radius, tides, waves, target height, or visibility.
- Not a navigation, surveying, aviation, or radio-link safety calculation.
- Very local topography can dominate the theoretical curvature limit.
Common mistakes
- Entering elevation above sea level when local height above the visible surface is needed.
- Treating the result as guaranteed visibility.
- Adding target height directly instead of calculating both horizon contributions.
- Confusing straight-line distance with surface distance.
Practical use cases
Observation planning
Estimate the curvature-limited baseline for an unobstructed view.
Geometry education
Apply the Pythagorean theorem to a tangent line and spherical radius.
Frequently asked questions
Does atmospheric refraction extend the horizon?
Often yes, but the amount varies with atmospheric conditions, so it is intentionally excluded.
What if the target also has height?
Calculate the horizon contribution for observer and target heights separately and add them as an approximation when appropriate.
Why use 6,371 km for Earth radius?
It is a common mean-radius approximation; local ellipsoidal curvature differs slightly.
Sources and review
- Compare Earth and the Moon — NASA Science. Accessed 2026-07-13.
Reviewed 2026-07-13.