Earth Curvature Calculator — How Far the Ground Falls

The Earth’s surface falls away from any tangent line as the square of distance. Enter a distance, and this calculator returns the geometric drop — and the figure surveyors actually use, softened by atmospheric refraction — plus the midpoint bulge between two points.

The Parabola That Isn’t Quite

Stand at one end of a straight line drawn tangent to the Earth and the surface recedes below it — slowly, then (being a square law) undeniably: drop ≈ d² ÷ (2R), with R = 6,371 km. The formula is the circle’s sagitta approximation, exact for the distances anyone walks (the true geometry replaces it beyond a few hundred kilometres, where the drop grows past the approximation’s manners). Its consequences are engineered everywhere sightlines matter: lake-spanning bridges curve their decks, sniper and artillery tables bend, the Laser Interferometer’s arms and particle accelerators’ rings are levelled against it, and the “eight inches per mile squared” of the surveyor’s rule is its imperial accent (0.203 m at one mile, the square doing the talking). The calculator adds the practitioner’s other two figures: refraction — the atmosphere bending light Earthward, conventionally modelled as a planet of 7/6 the radius — and the midpoint bulge, the surface’s rise between two points, which is the figure that actually obstructs a path drawn between them.

Worked Example: Ten Kilometres, Eight Metres

The defaults: at 10 km the geometric drop is (10,000)² ÷ (2 × 6,371,000) = 7.85 m — a two-storey house’s worth of planet curving away under a distance you can cycle in half an hour. Standard refraction softens the apparent figure to 6.73 m, and the midpoint bulge — the obstruction midway between two 10-km-apart points at equal height — is 1.96 m, the number a microwave link’s planner adds to the obstacle heights along the path (with the Fresnel clearance of the path-loss page on top). Scale the distance and watch the square rule the page: at 1 km the drop is 7.8 cm (a surveyor’s daily concern), at 100 km it is 785 m (a mountain’s), at 1,000 km the approximation itself must yield to the sphere’s true trigonometry — the calculator’s jurisdiction, stated plainly, is the local-to-regional range where the parabola figure is honest to centimetres.

Bedford, Bridges and the Levelled World

The drop’s measurability has a famous history: the Bedford Level experiments of the 1870s — boats and markers along a six-mile drainage canal in the Fens — were fought over by flat- and round-earth partisans precisely because this page’s six-mile figure (about 7.3 m geometric, refracted less) sits at the edge of what careful optics can certify from a wobbly boat, and refraction’s variability (the calculator’s 7/6 is a convention, not a constant of weather) let both sides claim their sightings. Surveyors settled the practical question by building the correction into their levels’ bookkeeping and moving on; the Transcontinental railroad, the Suez and Panama works, and every long aqueduct since are levelled against d² ÷ 2R as routine. The calculator inherits the surveyor’s convention openly — geometric truth, refraction adjustment, and the note that air is weather, not glass. Modern geodesy has, of course, retired the argument entirely: GNSS heights and satellite laser ranging measure the planet’s figure to centimetres as bookkeeping, and the drop this page computes is now the settled premise of every map, not a proposition anyone must defend from a boat.

The Longest Sightlines on Earth

Photographers and peak-baggers contest the world sightline record on this page’s arithmetic: the current marks exceed 400 km (Pic Gaspard to the Pyrenees’ Canigou, some 443 km in exceptional clarity), distances at which the geometric drop exceeds 12 km and only high summits, refraction’s generosity and the thinnest winter air permit the meeting of eye and peak at all. Every such record attempt begins with the calculator’s three numbers — the drop, the refracted drop, the bulge between the summits — and ends with a telephoto argument about which ridge is which. The square law that bends a surveyor’s level line by centimetres also writes the rules of the planet’s longest glances.

Level Datums and the Sea That Isn’t Flat Either

“Sea level” itself — the surveyor’s reference — is an equipotential of a lumpy gravity field (the geoid), departing from the calculator’s perfect sphere by up to a hundred metres globally, which is why ordnance datums are national heirlooms and GPS heights need geoid models to become heights above the sea. The calculator’s sphere is the regional truth the geoid wobbles around: over the distances of a link budget or a canal, the parabola on the mean radius is the operative curvature, and the geoid’s undulations are a correction the geodesist, not the link planner, carries. Naming the refinement’s address is the page’s honesty about its own round Earth.

Related Tools

More from the Physics Calculators hub: the Distance to Horizon Calculator — this drop’s mirror problem, from a height — and the Free Space Path Loss Calculator, whose links must clear this page’s bulge.

Frequently Asked Questions

Is the drop the same as how much of a distant object is hidden from view?

Not quite — hidden height depends on the observer’s elevation too: from a height h, the horizon itself dips, and an object beyond it is hidden by the bulge geometry measured from the observer’s tangent, which the horizon page of this hub handles in full (observer height in, visible range and hidden portion out). This page’s drop is the symmetric, observer-free quantity — the surface’s departure from a tangent at the stated point — which is what surveyors and link planners correct by. The two pages are a pair by design: curvature here, its consequence for eyes there.

Why does refraction change the figure at all?

Because air’s density — hence its refractive index — decreases with altitude, so a horizontal light ray bends gently Earthward, following the surface a little: distant objects appear raised, the drop appears reduced, and the effect is conventionally folded into an effective Earth radius of 7/6 R for optical work (4/3 R for radio, whose rays bend slightly more). The convention averages real weather; on days of strong temperature inversion the bending can double (looming, superior mirages) or vanish, which is why geodesy distrusts long optical sights and why this calculator labels its second figure “standard” rather than promising it.

What is the midpoint bulge used for in practice?

Path clearance: between two equal-height endpoints the surface’s highest intrusion into the straight line joining them is at the midpoint, of height d² ÷ (8R) — one quarter of the endpoint drop, as the parabola’s symmetry requires. Microwave and optical link designers add it (computed with the 4/3 radio Earth) to every terrain height along the path before checking Fresnel clearance; bridge and tunnel engineers meet it as the vertical curve their roadway must follow. It is the drop’s most professionally consequential form, which is why the calculator prints it alongside rather than leaving it as an exercise.

How did Eratosthenes use the curvature?

He measured its rate: the noon sun’s shadow angle differed by 7.2° between Syene and Alexandria, some 5,000 stades apart — from which the drop’s parent quantity, the circumference, fell out as 360 ÷ 7.2 × 5,000 = 250,000 stades, within a few percent of truth in the third century BC. This page’s R is his measurement refined by two millennia of geodesy (the modern mean radius 6,371 km, the figure the calculator embeds). Every drop the calculator returns is Eratosthenes’ angle integrated — the oldest great measurement in science, still on shift.

Does the formula work for other planets too?

Perfectly — only R changes: Mars (3,390 km) drops 1.88 times faster per kilometre squared (a Martian 10 km costs 14.8 m), the Moon (1,737 km) 3.67 times faster — a lunar horizon that crowds close and falls steeply, as Apollo’s crews’ photographs show. The calculator’s embedded R is Earth’s, stated in its working; the reader planning for Mars may scale any output by 6,371 ÷ R_planet. The square law itself is geometry’s, valid on any sphere — and on the gas giants’ cloud decks, mutatis mutandis.

How does this affect long-range shooting and artillery?

At battle ranges the target is measurably lower than a flat-earth line of sight would put it: over 10 km the drop is this page’s 7.85 m, an adjustment folded into artillery tables since the nineteenth century alongside the Coriolis terms of a rotating planet. Long-range marksmen meet the same correction in miniature (fractions of a metre at a kilometre’s-plus distances) inside their ballistic solvers, which embed the same d² ÷ 2R term the surveyor uses. The calculator’s regional jurisdiction covers all of it — the parabola does not care whether the line it corrects carries light, a shell’s aim, or a surveyor’s level bubble.

Why do some sources quote 8 inches per mile and others 7?

Both, about different quantities: 8 inches × (miles)² is the pure geometric drop (8.0 in at one mile, the calculator’s first figure in imperial dress: 0.203 m), while about 7 inches is the refraction-adjusted figure surveyors observe with (the calculator’s second). The inch of difference is the atmosphere’s standard bending, and arguments that quote one side against the other are usually this convention difference in disguise. The calculator prints both, labelled, so the reader can join such arguments armed — or decline them, better armed.