Earth Curvature Interactive Calculator

If you're working on a microwave link, radar system, or running a leveling survey over several kilometers, Earth's curvature isn't just textbook theory—it directly limits how far you can see or shoot a signal. This calculator lets you figure out horizon distance, how much height is hidden, true line-of-sight range, the height of the "bulge" at mid-span, dip angle, and the minimum height you need for visibility, all based on observer height, distance, and the Earth's radius. In RF telecom, marine navigation, or surveying, ignoring a 16-meter bulge on a 40 km path will cause costly mistakes. Scroll down for the core formulas, a real microwave path example, an explanation of atmospheric refraction, and practical questions on leveling and radar use.

What is Earth curvature?

Earth curvature is the amount the ground drops away from a straight line as you move horizontally. Since Earth is close to a sphere, two surface points are joined by a curve, but a straight line drawn from one to the other will end up above the ground by a measurable margin—that's the height "hidden" by the curve.

Simple Explanation

Picture standing on a huge ball—if you look straight out, the ground eventually drops away and objects disappear below the curve. The farther you look, the more gets blocked. That's why you can't see a ship's hull at long range, and why communication masts need height to close long links.

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Visual Diagram

Earth Curvature Calculator

How to Use This Calculator

1. Pick your calculation mode—horizon distance, hidden height, line-of-sight, bulge height, dip angle, or required target height.
2. Put in the numbers for your chosen case: observer height, any target heights, distance, or separate distances to each point as the form needs.
3. You can leave Earth radius at 6371 km (works for most jobs), or change it if you want equatorial or polar reference.
4. Hit "Calculate" to see the outcome.

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Equations & Variables

Simple Example

Observer height: 2 m. Earth radius: 6371 km.
Horizon distance: d = √(2 × 6,371,000 × 2) = √25,484,000 ≈ 5.05 km
Hidden height at 10 km: h = 10,000² / (2 × 6,371,000) ≈ 7.85 m
A 2-meter-tall person sees the horizon at roughly 5 km—at 10 km, almost 8 m of an object is blocked by the curvature already.

Use the formula below to calculate horizon distance.

Use the formula below to calculate hidden height (drop from curvature).

Use the formula below to calculate line-of-sight distance between two elevated points.

Use the formula below to calculate maximum bulge height.

Use the formula below to calculate dip angle to the horizon.

Theory & Practical Applications

Geometric Foundation of Earth Curvature

Earth curvature formulas come from basic spherical geometry. For most engineering, Earth’s average radius of 6,371 km is fine. If you're standing at height h above the surface, the furthest direct line-of-sight you get before losing view to the curvature is set by the straight tangent from your point out to the sphere. That’s a right triangle (hypotenuse is R + h, one leg is R, one is line of sight d). Apply Pythagoras: d² + R² = (R + h)², which then gives d = √(2Rh + h²). For observer heights much smaller than Earth's radius (almost always true), you can ignore the h² term: d ≈ √(2Rh) is accurate enough and quick to calculate.

The "drop" or hidden height h = d²/(2R) just flips the horizon formula and tells you how much lower the surface is at some distance d from your horizontal line of sight. This is critical for survey work—even across a few hundred meters, the effect isn’t huge, but by the time you’re into multiple kilometers, it’s measurable. Note that Earth isn’t a perfect sphere; it’s slightly squashed at the poles and fatter at the equator. Typical engineering projects can work with a mean radius. If you need high-precision over large distances, you’ll have to use the local effective radius or a published ellipsoid model. Using a ballpark radius might be off by about 0.3% at extreme latitudes or long distances.

Atmospheric Refraction Effects

Geometric curvature formulas only tell part of the story—light and radio rays don’t travel in perfectly straight lines in the lower atmosphere. Refraction usually bends rays slightly downward, extending the visible or radio horizon beyond the pure geometric case. The refraction coefficient, usually labeled k, often runs 0.13 - 0.16 for normal air (0.143 is a standard nominal value). This changes your “effective” Earth radius to R’ = R/(1-k), making the apparent curvature less and the range longer—by around 15-18%. So a calculated horizon of 11.3 km could become 13 km or so with typical refraction—it matters whenever you’re pushing range limits.

But k isn’t steady. Cold layers over water, dawn, or temperature inversions can cause k to jump, creating mirage effects and sometimes adding 50% or more to line-of-sight. In deserts or in strong sun, you might get less refraction—sometimes even the opposite, bending rays upward. At microwave frequencies, similar rules hold, but the effect can depend on weather, path height above ground, and the exact frequency. For most microwave work, an "effective radius" 4/3 larger than physical Earth (k = 0.25) is standard—not perfect, but practical and conservative when choosing tower heights or link clearance for varying conditions.

Applications in Telecommunications and RF Engineering

For wireless point-to-point links, just seeing the other antenna isn’t enough. You need to clear the first Fresnel zone—imagine an ellipsoid stretching between antennas, and if you block it, your link fades or fails. For example, a 6 GHz link (wavelength ~5 cm) over 40 km has a first Fresnel zone radius of about 14 m at the midpoint. Meanwhile, at that range, the "bulge" from Earth's curvature can be over 30 m high. So if your antennas are only a little above the ground, you’ll never get a clean path, no matter how perfectly aligned they are. Towers, terrain, and calculated margins all have to add up to clear both curvature and the Fresnel zone if you want a working link, especially above a few kilometers span.

Marine radar systems on ships typically sit 15–30 m above water; at 20 m, you might get about 16 km radar range before curvature blocks the view. If a small boat’s radar reflector is just 3 m above water, add another 6 km or so. In practice, the real range might be less because of clutter, absorption in heavy weather, and the small actual size (radar cross-section) of the target—not all of physics cares about geometric visibility.

Surveying and Geodetic Corrections

Leveling across long distances has to account for both curvature and refraction. The simple curvature correction h_c = d²/(2R) is reduced by the refraction effect (about 14% for k = 0.143), so you need to subtract that portion, not just plug in the flat curve value. Over a kilometer, expect about 67 mm total correction. Over 2 km, it’s over 250 mm. For highest-precision work, surveyors measure back and forth (reciprocal leveling) to cancel weather-related variation; single readings over several hundred meters need explicit corrections for both curvature and refraction.

For total stations shooting long slope lines, the curvature can add almost 2 meters of error over 5 km if ignored. GPS and GNSS receivers report ellipsoid heights, which are not the same as traditional leveling. Bottom line: if you want your numbers to add up across hills, valleys, or long spans, you need to know the curvature corrections and when to bring in models with more detail than a simple sphere.

Maritime Navigation and Vessel Detection

Navigation charts spell out lighthouse visibility assuming both the light's height and a “standard” observer on a vessel (usually around 4.5 m above water). At a typical 30 m lighthouse and 4.5 m observer, the geometric range is a bit over 27 km (about 15 nautical miles). Charts boost that by 8–10% if typical refraction is included. Light intensity and local weather can shrink real visibility to just 5 nautical miles even if the geometric conditions are there. AIS radio beacons—from vessel antennas about 10–15 m up—can reach shore stations over 30–35 km away as long as radio sensitivity and interference allow. Curvature sets an upper limit: if your antenna doesn’t clear the bulge, you won’t get through, no matter the transmitter power.

Worked Example: Microwave Link Path Analysis

Take a 23 GHz link between two buildings, 28.7 km apart, with a ridge at the midpoint. Antennas are at 45 m and 52 m, ridge is 38 m above average ground. Start with the geometric line of sight—using the horizon equations, total theoretical range is about 49.7 km, so there’s enough reach as far as direct, unobstructed line of sight.

The main issue is the midspan bulge. At 14.35 km from each end, the bulge is 16.2 m high. If you just stretched a string from one antenna to the other, that’s how much the ground “pushes up” at the midpoint. Now, figuring height at the midpoint along that straight line, and comparing with the ridge and bulge: there’s about 26.7 m of clearance over the ridge, well above needed first Fresnel clearance (around 8.2 m for this frequency and span).

With standard atmospheric refraction (k = 0.25, or “4/3 Earth radius” for RF), the effective bulge comes down to about 12.1 m, adding a few more meters of margin. You end up with a solid radio link and plenty of clearance. If you forget about bulge and just look at endpoint heights, you’d underestimate the clearance you actually need—curvature dominates the long path feasibility calculation, not local hills or antenna alignment. That's why you always check bulge and Fresnel on spans over 10–15 km, or you risk a failed installation.

Frequently Asked Questions

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📹 Video Walkthrough — How to Use This Calculator