How much does the Earth curve per mile?

At 1 mile, the Earth curves approximately 8 inches (0.67 feet or 0.2 meters). However, this is not a linear relationship — at 2 miles the drop is about 2.67 feet (not 16 inches), and at 10 miles it's about 66.7 feet. The popular "8 inches per mile squared" rule is a rough approximation that works at short distances but overestimates at long distances.

How far can I see before the Earth curves?

How far you can see depends on your height above sea level. Standing at the beach (eye height ~5.5 ft or 1.7 m), the horizon is about 2.9 miles (4.7 km) away. From a 100-foot lighthouse, you can see about 12.2 miles (19.6 km). From an airplane at 35,000 feet, the horizon is roughly 229 miles (368 km) away.

Do snipers calculate the curvature of the Earth?

For most sniper engagements (under 1,500 meters), Earth's curvature is negligible — the drop is about 3 inches at 1 km. However, at extreme long-range distances (2+ km), curvature becomes a factor alongside wind, Coriolis effect, air density, and bullet drop. The 2017 world-record shot at 3,540 meters would have required accounting for about 3.3 feet of curvature drop.

Earth Curvature Calculator

Distance to Target

Observer Height

Quick Presets

Atmospheric Refraction

Standard k = 7/6 model (~8% farther)

Horizon Distance

3.00miles

How far you can see from 6 ft above sea level

Target beyond horizon — 32.68 ft hidden

Curvature Metrics

Geometric (no refraction)

Curvature Drop

66.69ft

Hidden Height

32.68ft

Horizon Dip Angle

0.0434°

Midpoint Bulge

16.67ft

Reference Table

Curvature values at common distances (geometric)

Distance (miles)	Drop	Hidden (eye at 0)
1	0.6669 ft	0.6669 ft
2	2.67 ft	2.67 ft
5	16.67 ft	16.67 ft
10	66.69 ft	66.69 ft
20	266.75 ft	266.75 ft
50	1,667.2 ft	1,667.3 ft
100	6,668.4 ft	6,670.5 ft
200	26,669.4 ft	26,703.4 ft
500	166,497.5 ft	167,834.4 ft

What Is Earth's Curvature?

Why distant objects disappear bottom-first

Earth is an oblate spheroid with a mean radius of 6,371 km (3,959 miles). Because the surface curves away from any tangent line, distant objects progressively disappear from the bottom up. This is why a ship sailing away from shore appears to “sink” below the horizon — the hull vanishes before the mast.

The amount of curvature depends on distance: at 1 mile it's roughly 8 inches of drop, but the relationship is non-linear. At 10 miles the drop is about 66.7 feet, and at 100 miles it's over a mile. This calculator uses the exact geometric formula rather than the “8 inches per mile squared” approximation, which breaks down at long distances.

How Is Earth Curvature Calculated?

Geometric formulas for drop, horizon, and hidden height

All calculations model Earth as a sphere with radius R = 6,371 km. The observer stands at height h above sea level, looking at a target d away along the surface.

Horizon Distance

dₕ = R · arccos(R / (R + h))

Surface (arc) distance from the observer to the geometric horizon, derived from the angle subtended at Earth’s center.

Curvature Drop

drop = R · (1 − cos(d / R))

Vertical distance the surface drops below a horizontal plane, where d is the arc (surface) distance from the observer.

Hidden Height

hidden = R · (sec((d − dₕ) / R) − 1)

Minimum object height to be visible over the curvature. Zero if the target is closer than the horizon. Becomes infinite when the beyond-horizon arc reaches 90° — no object at that distance can clear the horizon.

Example: Looking across a 10-mile lake

1.Observer height: 6 ft (1.83 m) at the shoreline

2.Horizon distance: ≈ 3.0 miles (4.8 km)

3.Curvature drop at 10 miles: ≈ 66.7 ft (20.3 m)

4.Hidden height: ≈ 32.7 ft (10.0 m) — anything shorter on the far shore is invisible

Key Considerations

Important factors affecting curvature calculations

Atmospheric Refraction

Light bends as it passes through the atmosphere, typically extending the visible horizon by about 8%. The standard model uses an effective Earth radius of 7/6 × R (7,433 km). Actual refraction varies with temperature, pressure, and humidity.

“8 Inches per Mile Squared” Approximation

The popular approximation drop ≈ 8 × d² (inches, with d in miles) works well for short distances but increasingly overestimates beyond about 100 miles. This calculator uses the exact geometric formula instead.

Oblate Spheroid vs. Perfect Sphere

Earth is slightly flattened at the poles (equatorial radius 6,378 km, polar 6,357 km). This calculator uses the mean radius of 6,371 km. For most practical purposes the difference is negligible.

Terrain and Obstructions

These calculations assume a clear line of sight over a smooth surface (like open ocean). Hills, buildings, and other obstructions will block the view before curvature does.

Who Uses Earth Curvature Calculations?

Practical applications across industries

Surveyors & Engineers

Correcting for curvature in leveling surveys, bridge design, and long-distance construction projects.

Photographers

Predicting which landmarks are visible from a given elevation and how much is hidden by curvature.

Sailors & Navigators

Calculating the geographic range of lighthouses and determining when land becomes visible.

Radio & Telecom

Determining line-of-sight distances for antenna placement, microwave links, and radar coverage.

Aviation

Understanding horizon distance at cruising altitude for navigation and visual approach planning.

Educators & Students

Demonstrating Earth's spherical geometry and the observable effects of curvature.

Frequently Asked Questions

Common questions about Earth's curvature calculations

The curvature drop at surface distance d is calculated as: drop = R × (1 − cos(d/R)), where R is Earth's radius (6,371 km or 3,959 miles) and d is the arc (surface) distance. This gives the vertical distance the surface curves away below a horizontal plane. The horizon distance is dₕ = R × arccos(R/(R+h)). For the hidden height of a distant object beyond the horizon, the required object height to be visible is h = R × (sec(β) − 1), where β is the central angle from the horizon to the target.

The "8 inches per mile squared" rule (drop ≈ 8 × d² inches, with d in miles) is a simplified approximation of Earth's curvature. It works reasonably well for distances under about 100 miles but increasingly overestimates the drop at longer distances because it assumes a flat tangent plane rather than the actual spherical geometry. This calculator uses the exact geometric formula for accuracy at all distances.

Atmospheric refraction is the bending of light as it passes through layers of air at different temperatures and densities. This bending causes light to follow a slightly curved path rather than a straight line, effectively extending your visible horizon by about 8% under standard conditions. The standard refraction model uses an effective Earth radius of 7/6 × R (7,433 km instead of 6,371 km). Actual refraction varies with weather conditions.

From the Michigan shoreline (eye height ~5.5 ft), Chicago is about 60 miles away. The curvature would hide roughly 2,000 feet of the skyline. Since the tallest buildings (Willis Tower at 1,451 ft, including antenna at 1,729 ft) are shorter than that, they would be entirely below the geometric horizon. However, atmospheric refraction and temperature inversions can sometimes make the skyline partially visible — these are called "superior mirages" and are not uncommon over large bodies of water.

Higher elevation dramatically increases horizon distance. The relationship follows dₕ = R × arccos(R/(R+h)), which for moderate heights approximates to d ≈ √(2Rh). Horizon distance grows with the square root of height — doubling your height doesn't double the horizon, it increases it by about 41%. For example: at 6 feet the horizon is 3.0 miles, at 100 feet it's 12.2 miles, at 1,000 feet it's 38.7 miles, and at 35,000 feet it's 229 miles.

Curvature drop is how far the surface dips below a horizontal plane extending from the observer — it depends only on distance, not observer height. Hidden height is how much of a distant object is blocked by the curvature — it depends on both distance and observer height. If you're at sea level looking at a target 10 miles away, the hidden height equals the drop. But from a higher elevation, you can see over more of the curvature, so the hidden height decreases.

The horizon dip angle is how far below true horizontal the horizon appears. At sea level it's essentially zero. From higher elevations, the horizon dips noticeably: from a 100-foot lighthouse it dips about 0.18°, from an airplane at 35,000 feet it dips about 3.3°. This is why pilots see the horizon below the aircraft's level flight line, and it's used in celestial navigation to correct sextant readings.

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Last updated Apr 30, 2026