Earth Curvature Calculator
Enter a distance and observer height to instantly find how much Earth curves away (the drop), how much of a distant object is hidden below the horizon, how far away your horizon is, and how high you would need to stand to see a specific target. Switch between metric and imperial, add atmospheric refraction, and follow the step-by-step working below.
Formula
Worked example
Observer 1.7 m tall, 10 km from a target, standard atmosphere (k = 0.13): effective radius R' = 6,371,000 / 0.87 = 7,323,000 m. Drop = 7,323,000 - sqrt(7,323,000^2 - 10,000^2) = 6.84 m. Horizon = sqrt(1.7^2 + 2 x 7,323,000 x 1.7) = 4,988 m = ~4.99 km. Since the target is beyond the horizon, the bottom 1.73 m of the target is hidden.
What is Earth curvature and why does it matter?
Earth is not flat but an oblate spheroid with a mean radius of about 6,371 km. From any point on its surface, the ground curves away at a rate of roughly 7.8 centimetres per kilometre squared (the well-known "8 inches per mile" approximation). This curvature affects every long-distance observation: a lighthouse 30 km away may be hidden below the horizon, a mountain 100 km distant appears shorter than it is, and a surveyor's laser line over 10 km diverges from the true surface by about 7.85 m. Understanding these numbers matters in land surveying, navigation, photography, telecommunications tower placement, bridge engineering, and ballistics. This calculator gives you the exact curvature drop, hidden height, horizon distance, and required observer height for any distance and eye level, with optional atmospheric refraction.
Curvature drop, hidden height, and horizon distance explained
Curvature drop is the vertical distance Earth's surface falls below a perfectly straight horizontal line over a given distance. If you fired a perfectly horizontal laser beam, the ground beneath it would be 7.85 m lower after 10 km. Hidden height is how much of a target object (starting from its base) is hidden below the horizon when you look at it from a given eye height. If your horizon is 4.99 km away and the target is 10 km distant, the bottom portion of the target sinks below that horizon and becomes invisible without optical aid. Horizon distance is how far your horizon is from you given your eye height: standing on a beach 1.7 m above sea level gives a geometric horizon of about 4.65 km, or roughly 5 km with standard atmospheric refraction. These three quantities are linked by simple geometry based on the Pythagorean theorem applied to a sphere.
Atmospheric refraction and the k-factor
Light does not travel in a perfectly straight line through the atmosphere. Air density decreases with altitude, bending light rays slightly downward along Earth's curvature. The effect is expressed as a refraction coefficient k (also called the coefficient of refraction or the refraction factor). A k of 0 means no refraction (pure geometry in a vacuum). The standard value accepted for average atmospheric conditions is k = 0.13, which makes the effective Earth radius R' = R / (1 - k) approximately 7,323 km instead of 6,371 km. This extends the horizon and reduces the drop by about 14 percent compared to the geometric calculation. In stable, humid, or warm conditions refraction can increase to k = 0.17 or even k = 0.25, occasionally causing superior mirages in which objects well beyond the normal horizon become briefly visible. Surveyors and geodesists routinely apply the k = 0.13 correction to their field measurements.
How to use this calculator for real-world problems
For photographers and videographers planning a shot of a distant mountain, enter the distance and your tripod height in the observer eye field. The hidden height output tells you how much of the mountain's base is below your horizon. For surveyors checking whether a benchmark is visible, enter the distance and instrument height. If the required observer height output is greater than your instrument height, you cannot see the benchmark without raising the instrument. For ships at sea, enter the distance to a lighthouse and your deck height to find whether its light is visible above the horizon. The atmospheric refraction selector matters most at long distances, where the difference between k = 0 and k = 0.13 can be tens of metres. Select "none" for a conservative geometric lower bound and "standard" for the accepted surveying estimate.
Typical curvature drop values (no refraction, from sea level)
| Distance | Drop (geometric) | Approximate drop |
|---|---|---|
| 1 km | 0.078 m | 7.8 cm |
| 5 km | 1.96 m | ~2 m |
| 10 km | 7.85 m | ~8 m |
| 20 km | 31.4 m | ~31 m |
| 50 km | 196 m | ~0.2 km |
| 100 km | 785 m | ~0.8 km |
| 1 mi | 0.67 ft | ~8 in |
| 5 mi | 16.8 ft | ~17 ft |
| 10 mi | 67 ft | ~67 ft |
| 20 mi | 267 ft | ~267 ft |
Approximate curvature drop at standard distances using a geometric Earth radius of 6,371 km. Real values with atmospheric refraction (k = 0.13) are about 14% smaller.
Frequently asked questions
How much does Earth curve per kilometre?
The surface drops about 7.8 centimetres for the first kilometre and increases with the square of the distance. After 10 km the drop is about 7.85 m, and after 100 km it is about 785 m. The classic approximation is 8 inches per mile squared, or equivalently 0.078 m per km squared, which holds within a few percent for distances up to about 100 km.
What is the k-factor (atmospheric refraction coefficient)?
The k-factor quantifies how much the atmosphere bends light rays downward along Earth's curvature. A k of 0.13 is the standard surveying value for average conditions, making the effective Earth radius about 7,323 km and reducing the apparent curvature drop by about 14 percent. Higher k values (0.17 to 0.25) occur in warm, stable air near the surface and can extend visibility significantly. Setting k to 0 gives the pure geometric result with no atmosphere.
How far away is the horizon at sea level?
For an observer whose eyes are 1.7 m above sea level (standing on flat ground), the geometric horizon is about 4.65 km. With standard atmospheric refraction (k = 0.13) it extends to about 4.99 km. From a ship's bridge 10 m above sea level the horizon is about 11.3 km away (geometric) or about 12.1 km with standard refraction.
Why can I see objects that should be below the horizon?
Atmospheric refraction bends light downward, extending visibility beyond the geometric horizon. Under certain temperature inversion conditions, refraction can become strong enough to create a superior mirage, in which objects tens or even hundreds of kilometres away become briefly visible. This is why ships, islands, or even distant cities occasionally appear on the horizon when the geometry says they should be hidden. The k-factor can exceed 0.25 in such conditions.
Does Earth curvature prove or disprove flat Earth claims?
The curvature calculations in this tool are based on standard spherical geometry and confirmed by centuries of surveying, geodesy, satellite measurements, and GPS positioning systems. If Earth were flat, all the outputs in this calculator would be zero, yet real-world observations, such as ships disappearing hull-first below the horizon, agree closely with the spherical predictions. The curvature of about 8 cm per km squared is directly measurable by anyone with a theodolite, a laser level, and a long flat surface such as a canal.
What is the difference between drop and hidden height?
Drop is the geometric distance the surface falls below a horizontal line from the observer, regardless of what is at the far end. Hidden height is how much of a specific object at the far end is obscured from view because it is on the far side of the observer's horizon. Both depend on the distance, but hidden height also depends on the observer's eye height: standing taller moves your horizon further away and reduces the hidden portion of the target.
How accurate is the 8 inches per mile squared rule?
The approximation drop = (8/12) x d^2 (with d in miles, drop in feet) matches the exact formula to better than 1 percent for distances up to about 100 miles. Beyond that the approximation grows less accurate because it uses the small-angle linearisation. This calculator uses the exact Pythagorean formula: drop = R - sqrt(R^2 - d^2), which is accurate at any distance.