Earth Curvature Formula and How the Calculator Works
Earth's surface curves away from a level line of sight at a predictable rate. Because the drop is small compared with everyday sightlines, surveyors, sailors, and photographers commonly use the small-angle approximation h = d²/(2R), where d is the distance along the surface and R is Earth's mean radius (about 6,371 km / 3,959 mi). This calculator also finds how far the visible horizon is from a given eye height, using d = √(2Rh), and estimates how much of a distant target beyond that horizon is hidden by the curvature, with an optional correction for standard atmospheric refraction.
How the calculation works
Enter the distance to a target, the observer's eye height above the surface, and the unit each is measured in. The calculator squares the distance and divides by twice Earth's radius to estimate the curvature drop, then applies the same relationship in reverse — d = √(2Rh) — to find how far the geometric horizon lies from the entered eye height. When the target distance exceeds that horizon distance, the tool applies (d − d_horizon)²/(2R) to estimate how much of the target's height, measured from its base, is hidden below the horizon. Selecting "standard refraction" multiplies the true radius by 7/6 before running these formulas — the widely used approximation for how a typical atmosphere bends light and extends the visible range beyond the pure-geometry result.
Refraction and the horizon
- Refraction bends light downward as it passes through progressively denser air near the surface, letting you see slightly farther than pure geometry predicts.
- The common engineering approximation uses a refraction coefficient k ≈ 0.13-0.14, equivalent to multiplying Earth's true radius by 7/6 (effective radius ≈ 8,494 km / 5,278 mi) before applying the curvature and horizon formulas.
- Refraction strength changes with temperature gradient, humidity, and pressure near the ground, so treat the 7/6 correction as a typical long-run average rather than an exact value for any specific day or location.
Limits of the formula
The d²/(2R) drop formula is a small-angle approximation of the exact spherical relationship R(1 − cos(d/R)); it stays accurate to a fraction of a percent for distances up to a few hundred kilometers or miles, which covers essentially every practical horizon, surveying, or sightline question. It assumes a smooth spherical Earth with no terrain, buildings, or other obstacles in the way, and it only accounts for curvature — not obstructions blocking the view. For transcontinental distances or work requiring geodetic precision, use a full spherical or ellipsoidal Earth model instead of this approximation.