Flat vs. Round Earth Calculator

Enter a distance and your eye height to see how far away your horizon is, how far the surface drops below a level sightline, and how much of a distant object Earth's curvature would hide — the same horizon-dip formulas used in surveying and navigation.

Quick Facts

Curvature drop
h ≈ d² / (2R)
Over 1 mile the surface falls about 8 inches below a level line; the drop grows with distance squared.
Horizon distance
d = √((R+h)² − R²)
A 6 ft eye height gives a horizon around 3 miles away at sea level, before refraction.
Earth's mean radius
R ≈ 6,371 km (3,959 mi)
The IUGG mean radius used throughout this calculator; local terrain varies slightly.
Standard refraction
k ≈ 0.13 (≈ 7/6 R effective)
Light bends slightly around the curve, pushing the horizon a little farther than pure geometry alone.

Your Results

Calculated
Distance to Your Horizon
-
Farthest you can see before curvature blocks the view
Curvature Drop at Entered Distance
-
How far the surface falls below a level sightline
Hidden Height of the Object
-
Portion of the object's height blocked by curvature
Visible Height of the Object
-
Portion still visible above the horizon

Ready

Enter a distance, your eye height, and (optionally) the object's height, then press Calculate.

How the Flat vs. Round Earth Calculator Works

On a sphere, a level line of sight from your eye is tangent to the surface — it touches the globe at exactly one point (your horizon) and then rises above the ground beyond it. That means a distant object doesn't just shrink evenly with distance the way it would over a flat plane; once it passes beyond your horizon, its base disappears first while its upper portion stays visible a little longer. This calculator uses the same spherical-geometry formulas that surveyors, sailors, and lighthouse engineers use to compute your horizon distance, how far the surface has "dropped" below a level line at any distance, and how much of a distant object of known height would be hidden below that horizon.

Deriving the formulas

Picture a right triangle formed by Earth's center, your eye at height h above the surface, and the point where your line of sight just grazes the horizon. The hypotenuse is R + h (Earth's radius plus your eye height), one leg is R (the radius to the horizon point), and the remaining leg is the horizon distance d — so by the Pythagorean theorem, d = √((R + h)² − R²). The curvature drop — how far the surface has fallen below a level tangent line over a distance d — comes from the same triangle in reverse: drop = R(1 − cos(d/R)), which reduces to the familiar approximation drop ≈ d² / (2R) for distances that are small compared to Earth's radius. To find how much of a distant object is hidden, the calculator applies that same drop formula starting from your horizon point rather than from your eye, since your line of sight is itself tangent to the globe at that point.

Refraction, limits, and what this doesn't model

Earth's atmosphere bends light slightly downward as it travels near the surface, letting you see a bit farther than pure geometry predicts. The standard approximation multiplies the effective Earth radius by about 7/6 (equivalent to a refraction coefficient k ≈ 0.13) to account for this under typical conditions — toggle it off to see the plain geometric numbers instead. This model assumes a clear line of sight over water or flat terrain with no obstructions, and it doesn't account for temperature-inversion mirages, which can occasionally make refraction unusually strong or weak and let objects appear higher or lower than the standard formula predicts. It's also a small-angle approximation meant for line-of-sight distances up to a few hundred kilometers, not for antipodal-scale distances where the flat tangent-line picture breaks down entirely.

Frequently Asked Questions

Why do ships disappear hull-first instead of just shrinking into the distance?
On a globe, your line of sight is tangent to the curved surface, so it rises above the water the farther it travels past your horizon. A ship crossing that boundary has its lower hull blocked by the intervening curve first, while its masts and superstructure stay visible a bit longer. A truly flat surface has no such tangent-line effect; a ship would simply shrink with perspective and haze, never losing its hull while the top stayed sharp.
How far away is the horizon for an average person?
For a 6 ft (about 1.8 m) eye height at sea level, the geometric horizon is roughly 3 miles (4.8 km) away, growing to about 3.2 miles with standard atmospheric refraction included. Distance to the horizon scales with the square root of your height, so doubling your height only extends the horizon by about 41%, not double.
Does atmospheric refraction change these numbers a lot?
Under standard conditions, refraction extends the horizon and reduces the apparent curvature drop by roughly 14% compared to pure geometry — enough to matter for careful surveying but not enough to change the basic pattern. Unusual weather, such as a strong temperature inversion over cold water, can bend light far more than the standard model and occasionally make distant objects appear higher than expected; this calculator's standard refraction setting does not capture that situational effect.
What Earth radius does this calculator use?
It uses Earth's mean radius, about 6,371 km (3,959 mi), as adopted by the IUGG. Earth is actually a very slightly flattened spheroid, about 21 km wider at the equator than pole-to-pole, but that difference changes horizon and drop calculations by well under 1% at the distances this tool is meant for.

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