What does the earth curvature hide?
Type a distance and your eye height. The earth curvature calculator gives the curvature drop at that distance, the distance to your horizon, and how much of a far object the curve of the Earth hides.
- Hidden height
- 32.6814 ft
At 10 mi, the curve hides 32.6814 ft of a far object; the horizon is 2.99953 mi away.
- Curvature drop
- 66.6878 ft
- Distance to the horizon
- 2.99953 mi
Hidden height: 32.6814 ft. At 10 mi, the curve hides 32.6814 ft of a far object; the horizon is 2.99953 mi away.
How to calculate
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.
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.
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.
- 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
Each example is checked against the calculator on every build.
- Distance 1 mi, Eye height 0 ft, Earth radius 6,371 km gives Curvature drop 0.666876 ft, Hidden height 0.666876 ft, Distance to the horizon 0 mi.Source: NASA, Earth Fact Sheet (volumetric mean radius 6,371.000 km), https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
- Distance 10 mi, Eye height 6 ft, Earth radius 6,371 km gives Hidden height 32.6814 ft, Distance to the horizon 2.99953 mi.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
- Distance 1.864 mi, Eye height 6.562 ft, Earth radius 6,371 km gives Hidden height 0 ft, Distance to the horizon 3.13679 mi.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
How it works
The Earth is a sphere of radius R (6,371 km unless you change it). The distance d to the object is measured along the surface, and h is your eye height above the surface.
- Angle along the surface: θ = d ÷ R (radians).
- Curvature drop: the surface at distance d is R(sec θ − 1) below a level line from your feet. The page writes this as 2R sin²(θ/2) ÷ cos θ, the same value with no loss of digits at short distances.
- Distance to the horizon: the straight line from your eyes to the horizon, √(h(2R + h)). The horizon is at the angle θh = atan(√(h(2R + h)) ÷ R) along the surface.
- Hidden height: if θ ≤ θh, the object’s base is in view and nothing is hidden (0). Otherwise the hidden height is R(sec(θ − θh) − 1) = 2R sin²(φ/2) ÷ cos φ with φ = θ − θh. With h = 0 it equals the curvature drop.
Units: 1 mi = 1,609.344 m; 1 km = 1,000 m; 1 nmi = 1,852 m; 1 ft = 0.3048 m; 1 in = 0.0254 m; 1 cm = 0.01 m.
Output format. Every value shows 6 significant figures. The hidden height and the drop show in feet and the horizon distance in miles (meters and kilometers in Metric).
When there is no answer. A distance of one radius or more along the surface (θ ≥ 1 rad, 6,371 km on Earth), where a level line from your feet no longer meets the far surface in a useful way.
Assumptions
- A smooth sphere with no hills or waves, and straight sight lines (no atmospheric refraction).
- Distance 0.01 m to 5,000 km; eye height 0 to 100 km; radius 1,000 m to 100,000 km.
Worked examples by hand
One mile from the surface. θ = 1,609.344 ÷ 6,371,000 = 0.000252605 rad. Drop = 6,371,000 × (sec θ − 1) = 0.203264 m = 8.00251 in; with eyes at 0 the hidden height is the same, and the horizon is 0 away.
10 miles away, eyes 6 ft up (the default). Horizon √(1.8288 × (12,742,000 + 1.8288)) = 4,827.27 m (3.0 mi), at θh = 0.000757695 rad. θ = 16,093.44 ÷ 6,371,000 = 0.00252605 rad, so φ = 0.00176835 rad and the hidden height is 6,371,000 × (sec φ − 1) = 9.96128 m = 32.6814 ft.
3 km away, eyes 2 m up. Horizon √(2 × 12,742,002) = 5,048.17 m, beyond the object, so nothing is hidden.
Other questions people ask
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%.