Earth Curvature Calculator

Earth Curvature Calculator for the hidden height of a distant object and the curvature drop over any distance. It adds refraction, three Earth radii and the 8 inch per mile² check.

How the Earth Curvature Calculator Works

The Earth Curvature Calculator has two modes. Hidden Height of an Object is the default. It finds how much of a distant tower, ship, island or mountain sits behind the curve for an observer at a given eye height.

Curvature Drop Over a Distance answers the other common question. It gives how far the surface falls below a level line over any distance. It then checks that figure against the popular 8 inches per mile squared rule.

The folded Refraction, Radius and Units section sets the refraction coefficient, the Earth radius model and the result units. The default is pure geometry with the 6,371 km mean radius. Equatorial and polar radii are also available.

Reading the Results

The default case is a 6 ft eye height, an object 25 miles away and an object 1,000 ft tall. The Earth Curvature Calculator returns 322.8 ft hidden behind the curve, so the top 677.2 ft, or 67.7%, stays in view.

The second card shows that your own horizon is 3.00 miles away, so the object stands 22 miles past it. The horizon line sits 2.60′ below true eye level. The first card adds that the top would stay visible for another 16.72 miles.

The third card compares refraction settings. With a standard coefficient of 0.13, only 275.3 ft is hidden. The fourth card gives the plain curvature drop over 25 miles, 416.8 ft, against 416.7 ft from the 8 inch rule.

How Do You Calculate the Curvature of the Earth?

Draw a level line from a point on the sea surface. Over a distance D, the surface curves away below it. The drop is the gap between that level line and the surface. It follows from the angle θ = D ÷ R at the Earth’s center.

$$h_{drop} = \frac{R}{\cos(D / R)} – R$$

For distances much smaller than the Earth’s radius, this simplifies to the version most people quote. It is accurate to under 0.1% out to about 100 miles.

$$h_{drop} \approx \frac{D^2}{2R}$$

Putting R = 3,959 miles into D² ÷ 2R gives about 0.667 ft, or 8 inches, for one mile. Because the drop grows with distance squared, the rule is 8 inches times the miles squared, not 8 inches per mile.

Drop vs Hidden Height

The drop assumes your eye sits right on the surface. A real observer stands above it, so their sight line skims the surface at their own horizon before it carries on. Less of the far object is hidden than the drop suggests.

The Hidden Height mode first finds your horizon distance along the surface, s₁. It then applies the drop formula only to the stretch beyond it. That is why a 6 ft observer at 25 miles hides 322.8 ft of an object, not the full 416.8 ft drop.

$$h_{hidden} = \frac{R}{\cos\left( (D – s_1) / R \right)} – R$$

For an eye height h, s₁ is close to √(2Rh). A higher eye pushes s₁ farther out. That shortens the remaining span D − s₁, so less of the far object drops below the line of sight.

Earth Curvature per Km and per Mile

Earth curvature per km starts small. Over 1 km the surface drops 7.85 cm below a level line. Over 10 km it drops 7.85 m, a hundred times as much, because the drop grows with the square of distance.

Earth curvature per mile follows the same pattern. One mile gives 8 inches and two miles give 32 inches. Ten miles give 66.7 ft, and the average drop per mile keeps rising the farther you go.

Earth Curvature Chart

The chart below lists the exact drop from the formula, using the 6,371 km mean radius and no refraction. Inside the Earth Curvature Calculator, the same table also lists hidden height and visible height for your own eye height and object.

At 100 miles of curvature the drop is 6,671 ft, about 1.26 miles. At 500 miles it reaches 167,834 ft, nearly 32 miles. There the 8 inch rule falls about 0.7% short, which the calculator’s first card reports as the rule’s error.

How Much Does the Earth Curve in 10 Miles?

Over 10 miles the surface drops 66.7 ft below a level line from sea level. That is the figure for an eye right on the water. Stand at 6 ft on the shore and your horizon reaches 3 miles. Only 32.7 ft of a far object is then hidden at 10 miles.

With standard refraction the hidden part shrinks further. The gap between the bare drop and the real hidden height is a common mistake in curvature arguments. It is the reason the Earth Curvature Calculator asks for eye height.

Earth Curvature Calculator With Refraction

Air bends light gently downward, because it is denser near the ground. Light rays curve a little with the Earth, so distant objects appear higher than pure geometry predicts. Surveyors describe this with a refraction coefficient, k.

The calculator replaces R with an effective radius R ÷ (1 − k). The standard value is Gauss’s k = 0.13, shown in the menu as R × 1.15. Radio links use k = 0.25, the 4/3 Earth radius set in ITU-R Recommendation P.834. The Custom option accepts any value from −0.5 to 0.9.

Other sources use a slightly different standard. The Distance to Horizon Calculator, for example, uses a 7/6 radius, which equals k ≈ 0.143. Both are only typical values. A 2010 field study by Hirt and colleagues measured k from −4 to +16 near the ground on sunny summer days.

For the default case, refraction at 0.13 cuts the hidden part from 322.8 ft to 275.3 ft. Radio-style bending at 0.25 cuts it to 232.0 ft. Over cold water on a calm day, k can run much higher, which is why distant coasts sometimes appear to float into view.

Earth Curvature Calculator With Elevation

Observer elevation has the biggest effect on what is hidden. Every foot of eye height pushes your horizon farther out, by the square root of the height. The same 1,000 ft tower at 25 miles hides 322.8 ft from 6 ft, and far less from a hilltop.

Enter the eye height in feet, meters, kilometers or miles. For a beach observer, use eye level above the water. For a hilltop, use the ground elevation plus your eye height, both measured from the same water level as the target.

Earth Radius and Circumference

The Earth is slightly flattened, so no single radius fits every place. The calculator offers the three standard values listed in NASA’s Earth Fact Sheet. The mean radius suits most curvature questions. Use equatorial or polar only when you want to see how much the shape changes the answer.

So Earth’s radius is not 6,400 km, though that is a handy round figure. One Earth radius is 6,371 km on average. The circumference through the poles is about 40,008 km, close to 40,000 km because the meter was first defined from it.

One degree of the Earth’s curve spans about 111.2 km, or 69.1 miles, at the mean radius. One arcminute is about one nautical mile, which is how the nautical mile came about. The calculator’s drop mode shows this angle for any distance.

Who Figured Out the Curvature of the Earth?

Eratosthenes of Cyrene measured the Earth’s size around 240 BC. He compared the Sun’s angle at noon in Syene and Alexandria and found a difference of about 7.2°. He then scaled the distance between the towns up to a full circle.

His result was close to the modern value, depending on the length of the stadion. The Bucknell University astronomy notes work through his numbers. The same angle-to-distance idea sits behind the calculator’s surface angle card, which turns any distance into degrees of curve.

Can You See the Curvature of the Earth?

From the ground, no. At normal heights the horizon looks flat, because it dips less than a tenth of a degree. Many photos that seem to show a curved horizon are bent by wide-angle lenses rather than by the Earth.

The atmospheric optics researcher David Lynch studied this in Applied Optics in 2008. He found the curve becomes just visible at about 35,000 ft, with a cloud-free horizon and a field of view of at least 60°. That puts it within reach of pilots and airline passengers on a clear day.

So the curve is borderline from Everest and a little clearer at 40,000 ft. Being able to see detail along that far horizon is a separate question of optical resolution, which the Angular Resolution Calculator covers.

Curvature in Long-Range Shooting

Snipers and long-range shooters do not correct for Earth’s curvature on its own. Over 1,000 yards the surface drops only about 6.6 cm below a level line. The bullet’s own drop, many meters at that range, dwarfs it. The sight is also zeroed to the bullet’s path, not to a level line.

The Earth’s rotation matters more at extreme range. It shifts the bullet sideways and changes its apparent drop, effects the Coriolis Effect Calculator works out from latitude, direction and flight time.

Range to a distant target is usually read with a mil reticle rather than from a map. The Binoculars Range Calculator turns a target’s size and mil reading into distance.

Earth Curve Answers

These answers cover more of the curvature questions people ask. Each uses the same exact formulas as the Earth Curvature Calculator and the 6,371 km mean radius.

How far until the Earth starts to curve?

The Earth curves all the time, with no flat stretch. The drop is just too small to notice at short range. It reaches one inch at about 569 m, a little over a third of a mile, and one foot at about 1.2 miles.

A level line laid over a 100 m sports field shows a drop of under a millimeter. That is why builders and surveyors can treat short sites as flat, while long bridges and canals have to allow for the curve.

How do planes fly if the Earth is round?

An airplane holds its altitude above the ground, so it follows the curve on its own. Holding altitude means its nose drops very slowly as it flies. At 500 mph that adds up to about 7.2° per hour, or 0.12° per minute.

That rate is far below anything a pilot or passenger could feel. The autopilot simply keeps the altimeter steady, and the slow turn of the flight path happens without anyone needing to steer for it.

Can Everest be seen from India?

Yes, on clear days from high points near Darjeeling, such as Tiger Hill, roughly 170 km away. The combined horizons of a 2,590 m viewpoint and the 8,849 m summit reach over 500 km. At that range the curve hides none of the peak.

Haze, cloud and nearer ridges are the real limits. To test it, enter an 8,497 ft eye height, a 29,032 ft object and 106 miles. The Hidden Height mode reports the peak fully in view, with the base well inside the combined horizons.

Is there an Earth curvature calculator app?

You do not need to install one. The Earth Curvature Calculator runs in any phone or desktop browser, and its Share and Print buttons save a result for later. For an advanced Earth curvature calculator, open the folded section to set refraction and the radius model.

Results update as you type. The table below the cards also lists hidden height at standard distances, so you can read a range of cases without typing each one in.