How this instrument works
Curvature drop measures how far a point at some horizontal distance has fallen below a straight, flat plane drawn tangent to Earth's surface at your feet. Picture a perfectly flat table extending outward from where you stand; the real, curved ground sinks away from that table as distance grows, and drop = R − √(R² − d²) gives exactly how far down. The shape of the formula comes straight from the Pythagorean theorem applied to a circle: the tangent plane, the radius to your position, and the radius to the distant point form a right triangle, and the drop is simply the difference between the hypotenuse-length radius R and the leg √(R² − d²).
This is the exact secant-based drop, not the shortcut d²/(2R) rule that flat-Earth debunking videos often quote. That shortcut is a small-angle approximation of the same triangle, accurate to within a few percent out past 100 km but increasingly loose as distance grows, because it drops the higher-order terms that √(R² − d²) keeps. Near the extreme where distance approaches Earth's radius, the two diverge sharply — and the exact formula, not the approximation, is what a surveyor laying out a long baseline or a marine radio engineer computing line-of-sight range actually needs.
The number this instrument returns is pure geometry — it says nothing about whether light actually travels in a straight line to get there. Air density falls with altitude, which bends a sight line slightly downward and lets an observer see a little farther than flat geometry alone predicts; surveyors handle this by substituting an effective Earth radius, commonly about seven-sixths of the true one, into the same formula. Enter a distance equal to or beyond Earth's radius and the square root turns negative — a reminder that this tangent-plane construction only describes sight lines shorter than a quarter of the way around the planet, far beyond anything a human eye or a radio mast will ever need.
- Enter the Distance to observed object — the straight-line sight distance to whatever you're checking, in km or mi.
- Leave Earth radius at its 6,371 km default for Earth, or swap in the Moon's or Mars's radius to check curvature elsewhere.
- Read Curvature drop over that distance in meters or feet — how far the far point sits below a flat, tangent-line reference.
- Compare that drop to the height of the object's base, like a ship's waterline or a shoreline, to judge whether it would be hidden.
Worked example — a ship 50 km out to sea
With Distance to observed object set to 50 km (50,000 m) and Earth radius at the default 6,371 km (6,371,000 m), the formula gives R² − d² = 40,589,641,000,000 − 2,500,000,000 = 40,587,141,000,000 m², and the square root of that is 6,370,803.795 m. Subtracting from R leaves a Curvature drop over that distance of 196.204559428 m — about 196.2 m, or roughly the height of a 60-story building.
That is the classic ship-on-the-horizon observation: a large vessel's hull typically rises only fifteen to twenty meters above the waterline, so a 196-meter drop is far more than enough to sink the entire hull out of sight while its masts, still tall enough to poke above that drop, remain visible — the same effect early navigators used, centuries before satellites, to argue Earth must be curved.
Questions
Why isn't the formula just distance squared over twice the radius?
Because d²/(2R) is only an approximation of this instrument's exact formula, drop = R − √(R² − d²). Both trace back to the same right triangle, but d²/(2R) drops the smaller higher-order terms, so it tracks closely for short sight lines and drifts increasingly low as distance grows past a hundred kilometers or so. The instrument always computes the exact value.
Does the result include atmospheric refraction?
No — it is pure geometry, the drop below a flat tangent plane on an idealized sphere with nothing in the way. Real air bends light slightly downward, letting you see a little farther than this number alone suggests. Surveyors approximate that bending by plugging an effective Earth radius, often about seven-sixths of the true value, into the same formula rather than modeling the atmosphere directly.
What happens if I enter a distance equal to Earth's radius?
The square root inside the formula turns negative and the result stops being a real number, which is why the instrument flags distances at or beyond Earth's radius as invalid. Physically, a straight sight line that long would have to pass through the planet rather than skim its surface, far past anything a horizon calculation over open water or a radio link would ever call for.
Who actually needs a curvature-drop number like this?
Surveyors extending long baselines account for it when leveling; marine navigators use it to estimate how far away an approaching ship first clears the horizon; engineers planning microwave or line-of-sight radio links check it to confirm two towers can actually see each other over a curved Earth rather than through it.
Can this calculator be used for other planets or moons?
Yes — the geometry is identical for any sphere, so swapping the Earth radius field for the Moon's 1,737 km or Mars's 3,389 km gives the correct curvature drop for those bodies. Smaller radii produce a steeper drop at the same distance, which is part of why the Moon's horizon looks noticeably closer than Earth's to an observer standing on its surface.
Why do some online flat-Earth curvature calculators give a different number?
Most of those use the eight-inches-per-mile-squared shortcut, the small-distance approximation of this same formula, and many skip the atmospheric refraction correction surveyors normally apply. Both choices shift the result — sometimes by tens of meters at long range — which is why numbers copied from different calculators rarely match without checking which formula and which refraction assumption each one used.