# What does the earth curvature hide?

Works out how far the Earth’s surface curves away below a level line over a distance, the distance to the horizon from a height, and how much of a far object the curvature hides.

- Page: https://www.acalculator.org/physics/earth-curvature-calculator
- JSON spec: https://www.acalculator.org/physics/earth-curvature-calculator.json
- Version: 836fed07e2d7

## Default answer

Example with the default inputs (Distance 10 mi, Eye height 6 ft, Earth radius 6,371 km): At 10 mi, the curve hides 32.6814 ft of a far object; the horizon is 2.99953 mi away.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| d | Distance | How far away the object is, measured along the surface. |
| h | Eye height | How high your eyes (or the camera) are above the surface: 0 for the plain curvature drop. |
| R | Earth radius | The radius of the sphere: 6,371 km is Earth’s mean radius. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| hidden | Hidden height | How much of a far object is below your horizon: 0 if the object’s base is in view. |
| drop | Curvature drop | How far the surface at that distance is below a level line from your feet: R(sec θ − 1). |
| horizon | Distance to the horizon | The straight-line distance from your eyes to the horizon: √(h(2R + h)). |

## Method

θ = d ÷ R; drop = R(sec θ − 1); horizon = √(h(2R + h)); θh = atan(horizon ÷ R); hidden = R(sec(θ − θh) − 1) beyond the horizon, else 0.

## Assumptions

- The Earth is a smooth sphere of radius 6,371 km (NASA’s mean radius); the distance is measured along its surface.
- Light travels in straight lines: no atmospheric refraction, which in practice bends sight lines down and hides less.

## Worked examples

1. d = 1,609.344, h = 0, R = 6,371,000 gives drop = 0.203264, hidden = 0.203264, horizon = 0. Source: NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km), https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html.
2. d = 16,093.44, h = 1.8288, R = 6,371,000 gives hidden = 9.961284, horizon = 4,827.273863. Source: Wikipedia, Horizon: Distance to the horizon (d = √(2Rh + h²) for an observer at height h on a sphere of radius R), https://en.wikipedia.org/wiki/Horizon; NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km), https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html.
3. d = 3,000, h = 2, R = 6,371,000 gives hidden = 0, horizon = 5,048.168381. Source: Wikipedia, Horizon: Distance to the horizon (d = √(2Rh + h²) for an observer at height h on a sphere of radius R), https://en.wikipedia.org/wiki/Horizon.

## FAQ

### How much does the Earth curve per mile?

About 8 inches in the first mile: R(sec θ − 1) with R = 6,371 km gives 0.2033 m = 8.00 in. The drop grows with the square of the distance, so at 10 miles it is about 66.7 ft, not 80 in.

### Is the 8 inches per mile squared rule right?

It is a good shortcut for short distances: drop ≈ d² ÷ 2R, which is 8.0 in × miles². It gives the drop below a level line from your feet, not the hidden height, which also depends on how high your eyes are.

### How far away is the horizon?

From an eye height h, the straight-line distance to the horizon is √(h(2R + h)). From 6 ft (1.83 m) it is about 4,827 m, or 3.0 miles. From 100 m it is about 35.7 km.

### How much of a distant object is hidden?

Everything below your horizon line. The page works out the angle to the horizon, then the drop of the rest of the distance beyond it. From 6 ft, about 32.7 ft of an object 10 miles away is hidden.

### Why can I sometimes see more than the calculator says?

The air bends light slightly downward (refraction), so in practice less is hidden. A common rule of thumb models this with a larger radius, about 7/6 of the real one; you can type it under More options.

### What radius does the page use?

6,371 km, the Earth’s mean radius from NASA’s Earth fact sheet. The Earth is slightly flattened (6,378 km at the equator, 6,357 km at the poles), which changes these results by well under 1%.

## Sources

- NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km). https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html (retrieved 2026-10-03)
- Wikipedia, Horizon: Distance to the horizon (geometric distance d = √(2Rh + h²)). https://en.wikipedia.org/wiki/Horizon (retrieved 2026-10-03)
- NIST SP 811, Appendix B.8 (1 mi = 1,609.344 m; 1 ft = 0.3048 m; 1 nautical mile = 1,852 m). https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-03)