earth curvature calculator

Estimate Earth curvature drop over distance, plus an optional atmospheric refraction adjustment.

Distance along Earth's surface

Distance unit

Observer height above surface (meters)

Refraction coefficient k (0 to 0.5 typical, default 0.13)

Tip: Press Enter in any field to calculate.

What this Earth curvature calculator does

This tool estimates how much the Earth's surface drops away from a straight tangent line over a given distance. If you are comparing line-of-sight observations across water, long roads, or distant landmarks, curvature is often one of the first quantities you want to estimate.

You enter a distance, optionally include your observer height, and optionally account for atmospheric refraction. The calculator then reports geometric drop, approximate drop, apparent drop with refraction, and a simple hidden-height estimate.

How the calculation works

1) Geometric curvature drop

We model Earth as a sphere with mean radius R = 6,371,000 m. For arc distance d, exact drop from a tangent line is:

drop_exact = R × (1 − cos(d / R))

For shorter distances, a standard approximation is:

drop_approx ≈ d² / (2R)

The approximation is very close at practical distances used in many visibility checks.

2) Refraction-adjusted (apparent) drop

Light bends slightly in the atmosphere. A common engineering shortcut is to use an "effective Earth radius" by applying a refraction coefficient k:

R_eff = R / (1 − k)

Then compute apparent drop with R_eff in place of R. Typical near-surface value is around k = 0.13, but it changes with weather, temperature gradients, and pressure.

3) Horizon distance from observer height

Given observer height h, geometric horizon distance is:

horizon = √((R + h)² − R²)

How to interpret the results

Geometric drop (exact): baseline curvature with no atmospheric correction.
Approximation: fast check using the classic small-distance formula.
Apparent drop with refraction: what you may observe in standard atmospheric conditions.
Hidden height estimate: a simple estimate of how much of a distant object may be below line of sight.

Real-world viewing can differ from model output because terrain elevation, waves, lens distortion, mirage effects, and non-standard refraction can all dominate the scene.

Example

Suppose you enter 10 km, observer height 1.7 m, and k = 0.13. You should see curvature drop on the order of several meters. Refraction reduces apparent drop somewhat, and your small observer height reduces hidden amount only slightly at that distance.

Good use cases

Photography planning for long-distance landmarks
Marine and coastal line-of-sight checks
Educational demonstrations of Earth geometry
Sanity checks for observation claims at long range

Limitations and assumptions

Earth is not a perfect sphere; local radius varies slightly by latitude and terrain model.
Atmospheric refraction is simplified to one coefficient and can vary significantly moment-to-moment.
Hidden height is a simplified estimate, not a full optical ray-tracing solution.
Distances near antipodal scales are outside typical practical use for this simple model.

Bottom line

This calculator gives a quick, practical estimate of curvature effects and helps frame observations with a consistent geometric model. For high-precision surveying or geodesy, use professional tools that include ellipsoid geometry, terrain data, and full atmospheric profiles.