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The ideal geometric horizon from height h above a spherical Earth is D = √(h² + 2Rh), where R is Earth’s radius and all quantities use the same units. For Earth radius 6,371 km and heights in kilometres, the small-height approximation is D ≈ 112.88√h km. If both a UAV and a balloon or aircraft are elevated, calculate each endpoint’s horizon and add them to estimate the maximum unobstructed surface separation. For radio links, use the FAA approximation RLOS = 1.23√h1 + 1.23√h2 nautical miles, with heights in feet above mean sea level. These are planning limits, not guarantees: terrain, antenna height, atmospheric refraction, link budget and operating rules determine whether a real link or visual sightline works.
First decide which distance you need
People use “line-of-sight range” for several different measurements. Choosing the wrong one can produce a numerically precise answer to the wrong question.
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| Distance | What it measures | Use it for |
|---|---|---|
Geometric tangent distance (D) |
Straight-line distance from one elevated point to the tangent point on a spherical Earth | How far one UAV, balloon or aircraft can see before Earth curvature blocks a ray |
Surface distance (L) |
Earth-surface arc from the observer’s nadir to the tangent point | Mapping the horizon location on a chart or globe |
| Two-endpoint geometric horizon | Sum of the two endpoint tangent distances | Maximum ideal surface separation between two elevated platforms |
| Radio horizon | Approximate propagation range including standard atmospheric refraction | Initial VHF/UHF and similar communications planning |
| Slant range | Direct three-dimensional distance between two platforms | Radar, DME and telemetry readouts; it is not the same as ground distance |
A visual horizon calculation answers a geometry question. A radio-horizon calculation answers a propagation question. Neither one proves that a pilot may operate beyond visual line of sight or that a particular transmitter and receiver will close a link.
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Imagine the UAV at point P, height h above Earth’s surface, and the tangent point T where the line from the UAV just touches the spherical Earth. Earth’s centre O, the UAV and the tangent point form a right triangle. The radius OT is perpendicular to the tangent line PT.
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The distance from Earth’s centre to the UAV is R + h. Applying Pythagoras gives:
D² + R² = (R + h)²
Therefore the exact straight-line tangent distance is:
D = √((R + h)² − R²) = √(h² + 2Rh)
NASA’s horizon derivation uses this tangent geometry. It assumes a spherical, unobstructed Earth and a straight ray. The result is a chord-like line from the platform to the tangent point, not the distance you would measure along the ground.
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Surface distance to the tangent point
The central angle between the observer’s nadir and the tangent point is θ = arccos(R/(R + h)). Multiplying that angle by the radius gives the surface arc:
L = R arccos(R/(R + h))
For ordinary aviation and drone heights, D and L are close, but they are not interchangeable. Use D when a straight ray or slant geometry is required; use L when locating the horizon along Earth’s surface.
Why the square-root approximation works
When h is tiny compared with R, the h² term contributes little relative to 2Rh. The exact equation then becomes:
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D ≈ √(2Rh)
Using NASA’s Earth radius of 6,371 km and expressing height in kilometres produces:
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The square-root relationship is important operationally. Doubling altitude does not double the horizon; it multiplies the ideal distance by √2. Quadrupling altitude doubles it. This is why gaining altitude yields diminishing returns.
Calculate one platform’s geometric horizon
- Choose the distance type. Use the exact tangent equation for a precise geometric result, the approximation for quick planning, or the surface-arc equation when you need a ground location.
- Set one unit system. Convert height and Earth radius to the same units before substituting. With the NASA constant, use kilometres for both.
- Insert the height. For exact geometry, calculate
√(h² + 2Rh). For ordinary low-altitude work, calculate112.88 × √hin kilometres. - Convert after calculating. Convert kilometres to nautical miles or statute miles only at the end, so unit factors are not mixed inside the square root.
- Record the assumption. Label the result as an unobstructed spherical-Earth horizon and then screen it against terrain and the system’s link requirements.
Worked example: a 1-kilometre-high platform
With h = 1 km, the approximation gives D ≈ 112.88√1 = 112.88 km. The exact equation gives a value only slightly different because 1 km is very small compared with Earth’s 6,371-km radius. The figure describes the straight tangent distance from the platform to its geometric horizon; it does not promise a 112.88-km radio or visual connection over hills, buildings or haze.
Worked example: the general 400-foot small-UAS ceiling
Four hundred feet is approximately 0.12192 km. Substituting that height into the approximation gives a geometric horizon of about 39.4 km (21.3 nautical miles) for a single endpoint over a spherical Earth. The conversion is illustrative; the FAA operating limit and the horizon calculation answer different questions.
Calculate the horizon between a UAV and a balloon or aircraft
When both endpoints are above the surface, each has its own tangent distance. Under the same spherical-Earth assumptions, add them:
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For heights small relative to Earth’s radius, this simplifies to:
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Dmax ≈ √(2Rh1) + √(2Rh2)
This is the maximum ideal separation measured along the surface before Earth curvature blocks the line of sight. It is not the direct slant distance between the two vehicles. The actual straight-line distance between their antennas depends on their geometry and relative positions.
| Endpoint 1 | Endpoint 2 | How to calculate the geometric limit |
|---|---|---|
UAV at h1 |
Balloon at h2 |
Compute each exact tangent distance and add them |
UAV at h1 |
Aircraft at h2 |
Use the same two-height equation; aircraft type does not change the curvature geometry |
UAV at h1 |
Ground station at antenna height h2 |
Use the station antenna height above the local surface, not merely the building’s floor or mast base |
Example using the quick approximation
Suppose a UAV is at 1 km and a balloon is at 2 km. Using Earth radius 6,371 km:
- UAV horizon:
112.88√1 = 112.88 km. - Balloon horizon:
112.88√2 ≈ 159.63 km. - Ideal two-endpoint surface limit: approximately
272.51 km.
The 272.51-km figure is a curvature limit only. A link could fail much sooner because of antenna patterns, transmitter power, receiver sensitivity, Fresnel-zone obstruction, terrain, clouds or regulatory constraints.
FAA radio-horizon approximation
For radio planning, FAA engineering guidance gives a convenient two-height approximation:
RLOS (nautical miles) = 1.23√h1 + 1.23√h2
Use h1 and h2 in feet above mean sea level. The coefficient reflects an effective Earth radius of four-thirds the true radius, a conventional model for standard atmospheric refraction. It is an estimate for radio propagation, not a guaranteed communications distance.
| Input requirement | Correct treatment | Common mistake |
|---|---|---|
| Height unit | Feet for every endpoint | Putting metres or kilometres into the FAA expression |
| Output unit | Nautical miles | Calling the result kilometres without conversion |
| Reference level | Heights above mean sea level in the model | Mixing an elevation above sea level with an unadjusted local-ground height |
| Atmosphere | Effective 4/3-Earth-radius standard model | Treating the estimate as a guaranteed link or as visual sight |
Radio example at 400 feet
If both endpoints are 400 feet above mean sea level in the FAA expression, each contributes 1.23√400 = 24.6 nautical miles. The two-endpoint radio-horizon estimate is therefore 49.2 nautical miles, or about 91.1 km. This is larger than the approximately 78.8-km sum of two 400-foot geometric horizons because the radio model includes standard refraction through its effective-radius assumption.
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Do not interpret that difference as extra guaranteed coverage. FAA guidance notes that beyond the tangent point, VHF and higher-frequency signals do not ordinarily continue except during anomalous propagation. A real system still needs a link budget and a terrain profile.
Geometric horizon, radio horizon and visual line of sight are different
- Geometric line of sight: an unobstructed straight ray in a spherical-Earth model.
- Radio line of sight: a propagation estimate affected by frequency, refraction, diffraction, antenna height and system losses.
- Visual line of sight: whether the remote aircraft can be seen clearly enough by the observer to determine its attitude, location and direction without optical enhancement where rules require unaided sight.
- Regulatory permission: whether the operation is allowed in the relevant country, airspace, aircraft category and waiver framework.
A calculation can establish only the first item, or provide a first estimate for the second. It cannot grant permission or substitute for the operator’s visual-observer and airspace obligations.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Terrain, Earth shape and atmosphere: screening the ideal result
Terrain and obstacles
Hills, ridges, trees, towers and buildings can block a path long before the spherical horizon. Use a terrain-elevation profile between the two coordinates, include the height of each antenna above its local ground, and add a suitable clearance margin for the radio system’s Fresnel zone. A flat-Earth map distance is not a substitute for an elevation profile.
Local ground versus mean sea level
The geometric equations use height above the local spherical surface. The FAA radio expression specifies feet above mean sea level. If one endpoint is on a mountain, separate its terrain elevation from the antenna or vehicle height; otherwise the result can be off by the mountain’s elevation.
Earth model
Earth is not a perfect sphere. For short links, the spherical model is usually an adequate first pass, but geodetic calculations may use an ellipsoid and local datum. Use the same reference system for both endpoint coordinates and elevations when producing a terrain profile.
Atmospheric conditions
Temperature and pressure gradients bend radio rays. Standard refraction motivates the FAA four-thirds-Earth-radius approximation; super-refraction, ducting and other anomalous conditions can extend or reduce propagation unpredictably. Treat unusual long-distance reception as a propagation event, not as proof that the normal horizon is wrong.
Slant range is not ground range
The FAA Aeronautical Information Manual explains that distance-measuring equipment operates on a line-of-sight principle and reports slant range. Slant range is the direct three-dimensional separation between the aircraft and the station. When an aircraft is close to a station but high above it, the slant value can be substantially greater than the horizontal ground distance. Do not publish a DME or telemetry slant reading as a surface-distance measurement without converting it with the relevant heights and geometry.
Apply the calculation to a real communications link
- Define the endpoints. Record UAV antenna height, balloon or aircraft antenna height, terrain elevation, coordinates and the intended frequency.
- Calculate the geometric horizons. Use the exact equation when heights are large or precision matters; otherwise use the square-root approximation.
- Calculate the radio estimate. If planning in the FAA convention, convert both endpoint heights to feet and apply the 1.23 coefficient to obtain nautical miles.
- Check the surface path. Compare the proposed separation with the sum of the endpoint horizons and inspect terrain, structures and curvature along the path.
- Check the link budget. Include transmit power, cable loss, antenna gain and pattern, receiver sensitivity, polarization, frequency-dependent free-space loss, Fresnel clearance and interference.
- Test conservatively. Verify acquisition, throughput, latency and loss rate at the intended altitude and geometry. A receiver hearing a beacon is not necessarily receiving a usable command-and-control link.
- Check operating rules. Confirm visual-line-of-sight, altitude, airspace, observer and waiver requirements for the jurisdiction before flight.
Quick formulas and unit checklist
| Purpose | Formula | Required units |
|---|---|---|
| Exact one-endpoint tangent distance | D = √(h² + 2Rh) |
R and h identical |
| Small-height geometric approximation | D ≈ √(2Rh) |
R and h identical |
| NASA kilometre shortcut | D ≈ 112.88√h |
h in kilometres; result in kilometres |
| Surface arc to horizon | L = R arccos(R/(R+h)) |
R and h identical; angle in radians for calculation |
| Two-endpoint geometric limit | D1 + D2 |
Both endpoint distances in the same unit |
| FAA radio horizon | 1.23√h1 + 1.23√h2 |
Heights in feet; result in nautical miles |
For a spreadsheet, place Earth radius and each height in separate cells, use a square-root function for the exact or approximate formula, and label every output with its unit and model. Keep the exact and radio results in separate columns so a later user cannot mistake one for the other.
Regulatory context for U.S. small unmanned aircraft
FAA Part 107 guidance generally requires a small unmanned aircraft to remain within the remote pilot’s unaided visual line of sight and sets a 400-foot-above-ground limit, except where a stated exception or waiver applies. That rule does not alter the curvature calculation. It determines whether a proposed operation is lawful and how it must be conducted. A balloon or crewed aircraft may be visible at a distance that is geometrically possible while the UAV operation still fails an applicable visual-line-of-sight or airspace requirement.
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Mixing kilometres and metres
Putting h = 120 metres into a formula using R = 6,371 kilometres makes the height 1,000 times too large. Convert 120 m to 0.120 km first, or use an Earth radius in metres.
Using the FAA coefficient with metric heights
The 1.23 coefficient is paired with feet and nautical miles. For metric geometric planning, use √(2Rh) with a consistent radius or the 112.88-kilometre shortcut.
Adding heights instead of horizon distances
For two elevated endpoints, calculate each square-root horizon separately and then add the distances. Adding the altitudes produces no meaningful horizon range.
Confusing the tangent distance with the surface arc
The straight ray and the route along Earth’s surface are different quantities. Select D or L based on the question your map, sensor or operations plan must answer.
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Ignoring the lower endpoint
A high balloon does not make a low antenna effectively high. The lower endpoint’s horizon can be the limiting term, and a ridge near that endpoint can block the path even when the two-horizon sum looks large.
Treating a calculation as a link test
Horizon geometry says nothing about receiver sensitivity, interference, antenna orientation or data reliability. Use a radio link budget and field verification for those questions.
Quick Recap
A practical decision framework
- For a quick answer to “how far can a drone see from altitude?”, use the NASA kilometre shortcut after confirming the terrain is unobstructed.
- For the maximum ideal separation between a UAV and balloon, calculate both geometric horizons and add them.
- For a first radio-coverage estimate in U.S. aviation units, use the FAA expression with feet and nautical miles, then convert the result for your map.
- For an operational flight plan, replace the ideal result with terrain screening, a link budget, antenna details and the applicable visual-line-of-sight and altitude rules.
- For a measured DME or telemetry value, identify whether it is slant range before comparing it with a ground map.
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