What this calculator does
The Earth curvature calculator answers a specific question: over a given horizontal distance, how far does the surface drop away from a straight, level line drawn from the starting point? This is the "bulge" people mean when they talk about the curvature of the Earth affecting what is visible across open water, flat plains or long-distance photography.
It is a different question from how far you can see from a certain height, which is what a horizon-distance calculator answers. Curvature drop asks how much the ground itself curves away over a chosen distance, independent of how tall the observer is; sight distance to the horizon asks how far away the horizon appears given an observer's height above the surface. The two are related but answer different questions, so they are kept as separate calculators here.
The formula
Drop ≈ distance² ÷ (2 × Earth's radius). This is the standard small-angle approximation used for curvature-drop questions, accurate for the distances most people ask about, and it ignores atmospheric refraction, which in practice bends light slightly and makes objects appear a little less obscured by curvature than the pure geometry predicts. An exact geometric figure (radius − √(radius² − distance²)) is also shown for comparison, and the two match closely at normal sight distances.
| Term | Meaning |
|---|---|
| Drop | How far the Earth's surface falls below a straight, level line extended from the starting point, over the chosen distance. |
| Earth's radius | The mean radius of the Earth, about 6,371 km; a fixed physical constant, editable here for calculating curvature on other planets or moons. |
| Small-angle approximation | A simplification (distance² ÷ (2 × radius)) that is very accurate at the distances relevant to sightlines on Earth, avoiding the need for trigonometric functions. |
The inputs explained
| Field | What to enter |
|---|---|
| Distance (km) | The horizontal distance over which to measure the curvature drop. |
| Planet radius (Earth ≈ 6371 km) (km) | The radius of the planet or body; defaults to Earth's mean radius of 6,371 km. |
When to use it
Long-distance photography or sightlines
Across open water or flat terrain, working out the curvature drop over the distance to a distant landmark shows how much of it should be geometrically hidden below the horizon, before accounting for refraction or observer height.
Checking a flat-Earth claim
Curvature drop over a known distance, such as a bridge, causeway or stretch of coastline, gives a concrete, checkable number rather than an impression.
Surveying and engineering over long distances
Long bridges, tunnels and levelling surveys sometimes need a correction for Earth curvature once the distance involved is large enough for the drop to matter.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How much does the Earth curve over different distances?
The curvature drop over a range of common distances, using Earth's mean radius.
Questions
What is the formula for Earth's curvature?
The standard approximation is drop ≈ distance² ÷ (2 × radius), where drop and distance are in the same units and radius is Earth's mean radius, about 6,371 km. This is accurate for the ranges most sightline questions involve.
How much does the Earth curve over 10 km?
Using Earth's mean radius of 6,371 km, the surface drops about 7.85 m over a 10 km distance.
Is this the same as the distance to the horizon calculation?
No. This calculator finds how far the surface drops over a chosen distance. A separate distance to the horizon calculator finds how far away the horizon appears from a given observer height, which is a related but different geometric question.
Does this account for atmospheric refraction?
No, the main figures here are pure geometry. In reality, the atmosphere bends light slightly downward over long distances, which makes distant objects appear a little higher, and less obscured by curvature, than the geometric calculation alone predicts.
To work out how far away the horizon appears from a given height instead, see the distance to the horizon calculator.