Earth’s curvature can be calculated with surprisingly basic geometry: the Pythagorean theorem, a value for Earth’s radius, and your height above the surface. The core idea is that you, the surface, and the center of the Earth form a right triangle, and from that triangle you can derive how far away the horizon sits, how much a distant object is hidden below it, and how the surface curves away from you at any given distance. The formula works well enough for most practical purposes, but the real planet introduces complications worth knowing about.
The Fundamental Triangle
Picture yourself standing on a perfectly smooth, spherical Earth. Draw a line from the center of the planet to your feet, then extend that line up through your eyes. Draw another line from the center of the planet to the point on the surface where your line of sight just grazes the horizon. Those two lines are both radii, plus the extra bit of height where your eyes are. The line from your eyes to the horizon is tangent to the sphere, which means it meets the radius at the horizon at a right angle. You now have a right triangle.
The hypotenuse runs from the center of the Earth to your eyes, with a length of R + h, where R is Earth’s radius (about 6,371 km or 3,959 miles) and h is your eye height. One leg is R (from the center to the horizon point). The other leg is the distance from your eyes to the horizon. The Pythagorean theorem gives you (R + h)² = R² + d², which rearranges to d = √(2Rh + h²). Because your height is tiny compared to Earth’s radius, the h² term barely matters and you can simplify to d ≈ √(2Rh). That single expression is the workhorse behind almost every curvature calculation you will encounter.
Turning the Formula into Practical Numbers
If you are standing at the seashore with your eyes about 1.7 meters above the water, the horizon sits roughly 4.7 km (about 2.9 miles) away. Climb a 30-meter cliff and it jumps to about 19.5 km. Stand on the observation deck of a tall building 300 meters up and you can see roughly 62 km on a clear day. The relationship is not linear: doubling your height does not double the distance, it multiplies it by about 1.4 (the square root of 2).
People often want to know how much of a distant object is hidden behind the curve. The quick version: the amount of “drop” in the surface over a given distance grows with the square of that distance. Over 1 mile, the surface drops about 8 inches below a perfectly flat plane. Over 2 miles, it drops about 32 inches, not 16. Over 10 miles, roughly 66 feet of a distant building or ship would be hidden below the geometric horizon at eye level. These numbers assume a smooth sphere and no atmospheric bending of light, both of which matter in practice and are discussed below.
A useful rule of thumb for quick estimates: the horizon distance in miles is roughly 1.22 times the square root of your eye height in feet. In metric, the horizon distance in kilometers is about 3.57 times the square root of your eye height in meters. Neither formula is exact, but both get you within a few percent for everyday heights.
Measuring the Whole Planet With Shadows
The calculations above all depend on knowing Earth’s radius, and the first person to measure it with real precision was Eratosthenes of Alexandria, around 240 BCE. His method was elegant: he knew that at noon on the summer solstice, the sun was directly overhead in Syene (modern Aswan, Egypt), casting no shadow at the bottom of a well. At the same moment in Alexandria, roughly 800 km to the north, a vertical stick did cast a shadow. He measured the shadow angle at about 7.2 degrees, which is one-fiftieth of a full circle, and concluded that the distance between the two cities was one-fiftieth of Earth’s circumference. Multiplying gave him a circumference remarkably close to the modern value.
Modern students have replicated this approach with updated tools. A 2015 classroom experiment using two sites in Australia produced a circumference of 38,874 km, just 2.9% off the accepted equatorial value of roughly 40,075 km.1IOP Publishing (Physics Education). Modern replication of Eratosthenes’ measurement of the circumference of Earth The fact that a pair of sticks and some careful timing can get you within 3% of a satellite-derived answer says something about how accessible Earth-curvature calculations really are.
Why a Perfect Sphere Is Not Quite Right
Every formula above treats Earth as a perfect sphere, which it is not. Earth spins, and the centrifugal effect of that spin bulges the equator outward. The equatorial radius is about 6,378 km, while the polar radius is about 6,357 km, a difference of roughly 21 km. This makes the planet an oblate spheroid rather than a true sphere, and it means curvature is not the same everywhere. Walking north from the equator, the surface curves slightly more steeply (smaller radius of curvature) than walking east along the equator.
For most casual calculations, the difference is small enough to ignore. The sphere approximation introduces errors on the order of 0.3%, which is far less than the uncertainty introduced by atmospheric refraction or terrain. But for precision work in geodesy, surveying, and satellite navigation, the oblate shape matters. The standard reference model used by GPS and most mapping systems (WGS 84) defines an ellipsoid of revolution with specific equatorial radius and flattening parameters.
Some researchers have pushed the precision further still, modeling Earth as a triaxial ellipsoid with three slightly different axes rather than two. The equator itself is not perfectly circular; it bulges a bit more in some longitudes than others. A 2023 study in the Journal of Geodesy describes a physically motivated triaxial reference ellipsoid that accounts for both polar and equatorial flattening.2SpringerLink (Journal of Geodesy). A triaxial reference ellipsoid for the Earth For everyday curvature calculations, you will never need this level of detail, but it is worth knowing that the “shape of the Earth” question has layers of precision depending on your purpose.
How Atmospheric Refraction Bends the Numbers
If you have ever watched a ship disappear over the horizon and thought the distance seemed longer than your formula predicted, you were probably right. Light does not travel in perfectly straight lines through the atmosphere. Temperature and pressure gradients near the surface bend light downward, curving it slightly along the same direction as Earth’s surface. The effect is that you can typically see a bit farther than the geometric formula suggests, sometimes 5% to 15% farther under average conditions.
Surveyors and navigators have long accounted for this by applying a refraction correction. A common approximation replaces the true Earth radius R with an “effective” radius of about 7/6 × R, sometimes written as 7R/6 or expressed as a refraction coefficient of about 0.17. This effective radius stretches the horizon distance and reduces the apparent curvature drop at a given distance. The correction works reasonably well in temperate conditions but breaks down in extreme temperature inversions or when the air near the surface is much warmer or cooler than the air above it.
A detailed reanalysis of over 1,800 direct measurements of the horizon’s dip angle over the sea confirmed that the standard correction used in modern almanacs holds up well under typical conditions, but found that the dip behaves differently depending on whether the sea surface is warmer or cooler than the air above it.3Applied Optics. Refraction near the horizon—an empirical approach. Part 1: terrestrial refraction of the dip When the water is warmer than the air, refraction tends to increase and you can see farther; when the water is colder, the effect weakens or can even reverse. Under extreme conditions, light bends so strongly that objects well beyond the geometric horizon become visible, producing the mirage effect called “looming” that sailors have reported for centuries.
The practical upshot: if you are calculating curvature to figure out what you should be able to see from a given height, the geometric formula gives you a conservative baseline. On most days, refraction adds a few extra percent of visible distance. On unusual days, it can add a lot more or, rarely, subtract from it.
At What Altitude Can You Actually See Earth’s Curvature
There is a difference between calculating curvature on paper and perceiving it with your eyes. Many people assume you need to go to space to see the curve, but research on the subject suggests a lower threshold. A study using visual observations from high-altitude aircraft and photographs found that the minimum altitude at which curvature of the horizon becomes detectable to the naked eye is at or slightly below 35,000 feet (about 10,700 meters), provided the observer has a wide field of view of around 60 degrees and the horizon is mostly cloud-free.4PubMed. Visually discerning the curvature of the Earth
That altitude is right around commercial airline cruising height, which explains why some passengers notice a slight curve out the window and others do not. Airplane windows are small and restrict your field of view, which makes the curvature much harder to detect. If you could stand on top of the plane with an unobstructed panorama, the curve would be easier to spot. Below about 35,000 feet, any apparent curvature you see is likely either a psychological effect or distortion from a wide-angle camera lens.
Photographs can be misleading in both directions. A wide-angle or fisheye lens exaggerates curvature at any altitude, while a narrow telephoto lens can flatten it even from space. If you are evaluating a photo for evidence of curvature, the lens matters as much as the altitude. The most reliable visual test is a wide, unobstructed view of the horizon under clear conditions, where the curvature manifests as a subtle arc rather than a dramatic bend.
How Modern Geodesy Pins Down the Shape
The tools used to measure Earth’s shape today are a long way from sticks and shadows. The modern International Terrestrial Reference Frame, which is the coordinate system underlying GPS and essentially all global positioning, is built by combining observations from multiple space geodetic techniques: satellite-based positioning systems, satellite laser ranging, and very long baseline interferometry (VLBI), which uses radio telescopes to observe distant quasars.5Journal of Geophysical Research: Solid Earth. A Global Combination of Geodetic Techniques at the Observation Level: New Perspectives on the Terrestrial Reference Frame Each technique has different strengths: satellite systems cover the globe densely, laser ranging gives extremely precise distances to orbiting reflectors, and VLBI provides an absolute orientation reference tied to objects billions of light-years away.
These techniques are tied together at co-located sites where multiple instruments sit close enough to measure the precise physical offset between them, known as local-tie vectors.6Earth, Planets and Space. An effective approach for accurate estimation of VLBI–GNSS local-tie vectors The result is a reference frame that defines the shape and size of Earth to millimeter-level precision, tracking not only the overall curvature but also how the surface moves over time due to tectonic motion, glacial rebound, and tidal forces.
Satellite altimetry adds another dimension by bouncing radar pulses off the ocean surface to measure its height with centimeter-level accuracy. This technique has mapped the sea surface across most of the planet, including ice-covered regions of the Arctic that were previously inaccessible. ERS satellite altimeter data, for instance, has been used to construct a mean sea surface model of the Arctic Ocean between 60°N and 81.5°N.7Journal of Geophysical Research: Oceans. Sea surface height determination in the Arctic Ocean from ERS altimetry The ocean surface is not flat even after you account for waves; it rises and dips by tens of meters due to gravitational variations caused by underwater mountains, trenches, and differences in rock density. Mapping those undulations gives geodesists a detailed picture of the geoid, the surface of equal gravitational potential that curvature calculations at the highest precision levels are referenced to.
Common Mistakes People Make With Curvature Math
A few errors show up repeatedly when people try to apply curvature formulas, especially in online debates about what should or should not be visible across a body of water.
The most common is forgetting observer height. The drop formula (roughly 8 inches per mile squared) tells you how far the surface has fallen below a flat plane tangent to the Earth at your feet. But if you are looking at a distant object, the relevant question is not just how much the surface drops but how much of the object is hidden behind the bulge between you and it. That depends on both your eye height and the height of the target. Two people standing at sea level can each see just under 5 km to the horizon, which means they can see each other across a gap of nearly 10 km, even though the “drop” calculation at 10 km would suggest a substantial hidden portion. The bulge hides the middle of the distance, not the endpoints.
Another frequent error is applying the drop formula linearly. Because the hidden amount grows with the square of the distance, people sometimes calculate the drop for 1 mile and then multiply by 10 for 10 miles. The actual hidden amount at 10 miles is 100 times the 1-mile value, not 10 times. This quadratic relationship is the whole reason curvature is hard to notice at short distances but dramatic at long ones.
Refraction neglect is the third big source of error. On a warm day over cool water, you can often see objects that should be geometrically hidden. Without accounting for atmospheric bending, this looks like evidence against curvature, when it is actually evidence of how strongly the atmosphere refracts light near the surface. The correction is not trivial to calculate precisely, because it depends on local temperature gradients, humidity, and wind conditions.
Why Curvature Exists at All
It is easy to take for granted that planets are round, but not all objects in the solar system are. Small asteroids and comets are lumpy, irregular shapes. Roundness is a consequence of gravity: once a body accumulates enough mass, its own gravitational pull is strong enough to overcome the structural rigidity of rock and ice, pulling everything toward the center and producing a roughly spherical shape. Analysis of solar system bodies suggests that objects with a radius below about 160 km tend to be non-spheroidal, while those above about 450 km are consistently spheroidal, with a transitional zone in between.8arXiv. The Minimum Mass of Planets, Dwarf Planets, and Planetary-scale Satellites
Earth, with a radius of over 6,300 km, is far past that threshold. Its curvature is gentle enough to be imperceptible at human scales but relentless enough to hide a city skyline at 50 miles. The same physics applies everywhere in the solar system. Mars curves more sharply (smaller radius), the Moon more sharply still, and Jupiter curves far more gently. If you wanted to calculate horizon distances on any of those bodies, you would use the same Pythagorean formula; you would just substitute a different radius and, in the case of bodies with atmospheres, a different refraction correction. The geometry is universal. The specific numbers change with every world.