Free Online Earth Curvature Calculator
Estimate horizon distance and curvature drop over a distance.
Eye height, camera height, or the top of a tower.
Up to 2,000 km. Beyond that the tangent-plane model is no longer meaningful.
Uses a refraction coefficient of 0.13, which increases the effective Earth radius by about 15%.
The hidden height is the part of a target that lies below the observer's line of sight, assuming a smooth spherical surface and no terrain in between.
Curvature drop by distance
How far the surface falls away from a horizontal tangent over common distances.
| Distance | Curvature drop | Drop in feet |
|---|---|---|
| Calculate to populate this table. | ||
Values use the current refraction setting. The approximation is most reliable over short and medium distances; over hundreds of kilometres the curvature of the path and atmospheric variability become significant.
About the Earth Curvature Calculator
Earth's surface curves away from a horizontal line of sight, which is why distant ships disappear hull-first and why a tall tower can be seen from farther away than a person standing on the beach. This calculator relates three quantities: the height of an observer, the distance to the visible horizon, and the amount by which the surface drops over a chosen distance. It also estimates how much of a distant target is hidden below the observer's tangent line. The model assumes a smooth sphere and optionally accounts for the average bending of light in the atmosphere.
Geometry of the horizon
For an observer at height h above a sphere of radius R, the line of sight to the horizon is tangent to the surface. The distance along the surface to that tangent point is closely approximated by d ≈ √(2Rh), and the exact spherical form adds an h² term: d = √(2Rh + h²). For everyday heights the difference is negligible. The dip angle, the angle between the horizontal and the tangent line, is arccos(R / (R + h)). A higher observer sees farther and has a larger dip angle.
Curvature drop
Over a surface distance d, the surface falls below the tangent line by R − √(R² − d²). For distances much smaller than the radius, this is approximately d² / (2R). The familiar rule of thumb that the drop is about 8 inches per mile squared comes from this approximation. The drop grows quadratically, so doubling the distance quadruples the drop. At 10 kilometres the drop is roughly 7.8 metres on a sphere without refraction, and at 50 kilometres it is about 196 metres.
Atmospheric refraction
Air density decreases with altitude, so light travelling nearly horizontally bends downward. This makes objects appear slightly higher than their geometric position and extends the visible horizon. A standard refraction coefficient k ≈ 0.13 is often used, equivalent to replacing Earth's radius with an effective radius R / (1 − k), about 1.15 times larger. Refraction varies with temperature, pressure, and the vertical temperature gradient, and abnormal conditions can produce mirages, looming, or a horizon that appears higher or lower than predicted. The checkbox lets you compare the geometric and refracted results.
What the hidden-height result means
If the curvature drop over the target distance is greater than the observer's height, the base of the target lies below the tangent line and is hidden. The difference is the hidden height. A target taller than that value can still be seen above the horizon. This calculation ignores terrain, waves, buildings, and the target's own elevation above sea level, so treat it as a geometric baseline rather than a precise visibility forecast. For radio paths and surveying, use a proper terrain profile and a refractive index model.
Frequently asked questions
Why does the calculator use 6,371 kilometres for Earth's radius?
That is the mean radius of a sphere with the same volume as the ellipsoid. Earth is slightly flattened, so the local radius varies from about 6,335 kilometres at the poles to 6,378 kilometres at the equator.
Is the eight-inches-per-mile rule accurate?
It is a useful approximation for small distances and ignores refraction. The exact spherical drop at one mile is about 20 centimetres, or 7.9 inches. At longer distances the approximation drifts, so use the exact formula for precise work.
Can I see farther from a mountain?
Yes. Horizon distance grows with the square root of observer height, so a tenfold increase in height extends the horizon by about 3.16 times. A 1,000-metre summit can see more than 110 kilometres to a sea-level horizon under standard refraction.