The 8-inches-per-mile-squared rule
The vertical amount the Earth curves away from a straight line of sight is roughly 8 inches times the distance in miles squared. Because it uses the square, the drop is small up close and grows fast: double the distance and the drop quadruples. The precise form is drop = distance² ÷ (2 × Earth radius), with a radius of 3,959 miles.
How fast the drop builds with distance
| Distance | Curvature drop | Rough comparison |
|---|---|---|
| 1 mile | 8 in (0.67 ft) | knee height |
| 2 miles | 2.7 ft | a tabletop |
| 5 miles | 16.7 ft | a two-story house |
| 10 miles | 66.7 ft | a six-story building |
| 20 miles | 267 ft | a 25-story tower |
| 50 miles | 1,667 ft | taller than most skyscrapers |
This is the geometric hidden height from a straight sightline. What you can actually see also depends on your own eye height, which lifts the horizon.
Horizon distance depends on how high you stand
| Eye height | Horizon (geometric) | With refraction |
|---|---|---|
| 5 ft (eye level) | 2.7 mi | 2.9 mi |
| 10 ft | 3.9 mi | 4.2 mi |
| 100 ft | 12.3 mi | 13.2 mi |
| 1,000 ft | 38.9 mi | 42.0 mi |
| 35,000 ft (airliner) | 233 mi | 251 mi |
Height helps with diminishing returns: doubling your elevation extends the horizon by about 41 percent, not double, because the distance scales with the square root of height.
Two things people mix up
- Drop is not the whole story. The curvature drop is measured from a flat sightline. Whether an object is actually hidden also depends on your eye height, which raises the horizon and lets you see farther.
- Refraction adds about 8 percent. Air bends light toward the ground, so real horizons sit slightly beyond the pure-geometry figure. Surveyors handle this by using an effective Earth radius of 7/6 the true value.
Common questions
How much does the Earth curve per mile?
About 8 inches over the first mile. But it is not a steady 8 inches each mile: the drop grows with the square of the distance, so at 2 miles it is roughly 32 inches, and at 10 miles it is about 66 feet.
Why do distant ships disappear from the bottom up?
As a ship sails away, more of its hull sinks below the geometric horizon while the mast stays visible. On a flat surface the whole ship would just shrink evenly with distance. The hull vanishing first is a direct sign of curvature.
How far is the horizon from eye level?
For a standing adult with eyes about 5 to 6 feet up, the horizon is roughly 3 miles away. Atmospheric refraction bends light slightly and adds about 8 percent, so you usually see a touch farther than the geometry alone predicts.
