Earth Curvature Calculator
Find how much the Earth curves away over any distance and how much of a distant ship, island or skyline is hidden behind the bulge from your eye height. Add a refraction coefficient to see how the atmosphere bends light and lets you see a bit farther.
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Eye level above the water or ground, e.g. 6 ft standing on a beach
Same unit as eye height: a tower, ship, island or building
0 = pure geometry, 0.13 = standard atmosphere, ~0.17 over cool water
How to use the Earth Curvature Calculator
- Enter the distance to the target and pick a unit.
- Enter your eye height above the water or ground.
- Add the target height (optional) to see how much of it shows.
- Set refraction: 0 for pure geometry, 0.13 for a typical atmosphere.
- Press Calculate to see hidden height, drop, horizon distance, the diagram and the table.
Earth curvature formulas
- Curvature drop over distance d:
drop = R ÷ cos(d ÷ R) − R ≈ d² ÷ 2R. - Horizon distance from eye height h:
d₀ = R · arccos(R ÷ (R + h)) ≈ √(2Rh). - Hidden height of a target at distance d:
hidden = R ÷ cos((d − d₀) ÷ R) − Rwhen d > d₀, otherwise 0. - Refraction: replace R with the effective radius
R′ = R ÷ (1 − k). - Bulge at the midpoint between two points at sea level:
R · (1 − cos(d ÷ 2R)) ≈ d² ÷ 8R.
The calculator uses the mean Earth radius of 6,371 km (3,959 miles). The real Earth is slightly flattened, so the local radius varies by about ±0.3%, which changes results by the same small fraction.
Curvature drop reference table
| Distance | Drop (no refraction) | Hidden from 6 ft eye height | Hidden with k = 0.13 |
|---|---|---|---|
| 1 mi (1.6 km) | 8 in (0.20 m) | 0 | 0 |
| 3 mi (4.8 km) | 6.0 ft (1.8 m) | 0 | 0 |
| 5 mi (8 km) | 16.7 ft (5.1 m) | 2.7 ft (0.8 m) | 1.8 ft (0.6 m) |
| 10 mi (16 km) | 66.7 ft (20.3 m) | 32.7 ft (10.0 m) | 26.7 ft (8.1 m) |
| 20 mi (32 km) | 267 ft (81 m) | 193 ft (59 m) | 163 ft (50 m) |
| 50 mi (80 km) | 1,667 ft (508 m) | 1,473 ft (449 m) | 1,270 ft (387 m) |
Why you can often see "too far"
Light bends slightly downward as it passes through air that gets thinner with height, so you see around the curve a little. On cold water under warm air the bending can be much stronger and can even produce mirages that lift or stretch distant coastlines. That is why observations of far-away objects should be compared with a refraction value, not just the bare geometry, and why the same view can change from day to day.
Frequently asked questions
How much does the Earth curve per mile?
The surface drops about 8 inches (20 cm) below a level line after 1 mile, 32 inches after 2 miles and 16 feet after 5 miles. The drop grows with the square of distance: drop ≈ d² ÷ 2R, which for miles gives the familiar 8 inches × miles² rule.
Is the 8 inches per mile squared rule correct?
It is a good approximation of the drop from a level line for an observer at ground level over short to moderate distances (under a few hundred miles). It does not account for your eye height, which hides less, or for refraction. For seeing distant objects, hidden height is the more useful number.
What is the difference between drop and hidden height?
Drop is how far the surface falls below a horizontal line starting at ground level. Hidden height is how much of a distant object is blocked by the bulge when viewed from your actual eye height. Because your eyes are above the ground, hidden height is always less than the drop.
How is hidden height calculated?
First find your horizon distance d₀ = R · arccos(R ÷ (R + h)). If the target is beyond the horizon, hidden height = R ÷ cos((d − d₀) ÷ R) − R, where R is the Earth radius (6,371 km) and d is the distance along the surface. With refraction, R is replaced by the larger effective radius R ÷ (1 − k).
What refraction coefficient should I use?
A standard atmosphere has k ≈ 0.13 (about 1/7.5), which makes the Earth behave like a sphere about 15% larger. Over cool water k can reach 0.17 or more; over hot ground it can be close to 0 or even negative. Use 0 for pure geometry.
How far is the horizon from 6 feet?
From an eye height of 6 feet (1.83 m) the geometric horizon is about 3.0 miles (4.8 km) away, or about 3.2 miles (5.2 km) with standard refraction. Doubling your height increases horizon distance by about 41%.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.