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Direct answer: For a UAV, balloon, or aircraft at height h above a spherical Earth, the exact straight-line distance to the geometric horizon is D = √(h² + 2Rh), where R is Earth’s radius and both values use the same units. With R = 6,371 km, the practical approximation is D ≈ 112.88√h kilometres when h is in kilometres.
When both ends of a link are elevated, calculate each endpoint’s horizon and add them for the maximum unobstructed geometric line-of-sight distance. For a radio link, the FAA approximation is RLOS = 1.23√h₁ + 1.23√h₂ nautical miles, with heights in feet above mean sea level. That radio estimate uses an effective 4/3 Earth radius and is not a guaranteed communications range.
What “line-of-sight range” means
Altitude alone does not produce one universal range number. The answer depends on which distance you need and what kind of signal or observation is involved.
- Geometric horizon distance: the straight ray from an elevated endpoint that just touches Earth’s surface. This is the maximum ideal distance before the spherical Earth blocks the ray.
- Surface or ground distance: the arc measured along Earth’s surface from the observer’s nadir to the tangent point. It is shorter than the straight-line distance through the air.
- Radio horizon: an engineering estimate that allows for ordinary atmospheric refraction. It can exceed the vacuum geometric result, but terrain, antenna performance, and atmospheric conditions still determine whether a usable link exists.
- Visual line of sight: whether a person or camera can actually see the other object. Earth curvature is only one obstruction; hills, buildings, haze, cloud, lighting, and target size also matter.
The calculations below give an ideal first screen. They do not replace a terrain profile, an aviation assessment, or a radio link budget.
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Inputs, reference levels, and units
Use the height of each endpoint above the Earth reference surface used by your calculation. The equations are simplest when heights are measured above mean sea level and Earth is represented by a sphere. For a UAV, operational altitude is often reported above ground level (AGL), while maps and aircraft systems may provide mean-sea-level (MSL) altitude. Convert an AGL value to an MSL height by adding the local ground elevation before applying a two-aircraft or radio-horizon calculation.
Keep the radius and height in identical units for the exact equation. If R is in kilometres, enter h in kilometres; if R is in metres, enter h in metres. The FAA radio approximation is different: it specifically uses feet for both endpoint heights and returns nautical miles.
| Quantity | Required input | Result | Use it when |
|---|---|---|---|
| Exact geometric slant distance | R and h in the same units |
√(h² + 2Rh) |
You need the tangent ray length from one elevated endpoint |
| Surface distance to tangent point | R and h in the same units |
R arccos(R/(R+h)) |
You need distance measured over Earth’s surface |
| Two-endpoint geometric limit | Two endpoint heights in the same units | D₁ + D₂ |
You want the maximum ideal separation between two elevated sites |
| FAA radio-horizon estimate | h₁ and h₂ in feet MSL |
1.23√h₁ + 1.23√h₂ nautical miles |
You are making a first-order VHF-or-higher radio-coverage estimate |
Exact geometric horizon from one altitude
Draw a line from the UAV to the horizon point. At the horizon, that line is tangent to Earth, so it is perpendicular to the radius drawn to the tangent point. The observer, Earth’s centre, and tangent point form a right triangle. The observer is R + h from the centre, one triangle leg is R, and the tangent line is the horizon distance D.
Applying the Pythagorean theorem gives:
D² + R² = (R + h)²
Exact straight-line horizon distance:
D = √((R + h)² − R²) = √(h² + 2Rh)
This is a straight-line, or slant, distance through the air from the endpoint to the tangent point. It is not the distance that a map measures along the ground.
Surface distance to the tangent point
The central angle between the point directly below the observer and the tangent point is θ = arccos(R/(R+h)). Multiplying that angle by the radius gives the surface arc:
Ground-arc distance: L = R arccos(R/(R+h))
For low altitudes compared with Earth’s radius, D and L are close. At aircraft or balloon altitudes of several kilometres, the distinction becomes more useful, especially when you are comparing a slant-range instrument reading with a map distance.
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The small-height approximation
Because ordinary UAV and balloon heights are tiny compared with Earth’s radius, the h² term is usually much smaller than 2Rh. Dropping that term gives:
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D ≈ √(2Rh)
Using NASA’s Earth radius of 6,371 km:
D ≈ 112.88√h kilometres, with h in kilometres.
The square-root relationship is the key planning rule. Doubling altitude does not double horizon range; it multiplies the range by √2. Quadrupling altitude doubles the ideal horizon distance.
| Height above surface | Approximate geometric horizon | Interpretation |
|---|---|---|
| 0.01 km (10 m) | About 11.29 km | Low camera or antenna above the surface |
| 0.1 km (100 m) | About 35.70 km | Small-UAS-scale altitude in perfectly open terrain |
| 1 km | About 112.88 km | Illustrative balloon or aircraft endpoint |
| 10 km | About 356.95 km | High-altitude aircraft endpoint; approximation remains close but atmospheric and operational factors dominate |
These are geometric distances from a single endpoint. They do not claim that a camera can resolve an object at that distance or that a radio can deliver a usable data rate.
Calculating the maximum range between two elevated endpoints
If a UAV and a balloon or aircraft are both above the surface, each one can see to its own horizon. The maximum ideal separation occurs when their tangent points meet, so add the two endpoint horizons:
Dmax = √(h₁² + 2Rh₁) + √(h₂² + 2Rh₂)
For ordinary heights, the faster version is:
Dmax ≈ √(2Rh₁) + √(2Rh₂)
This is a maximum separation under the spherical-Earth, unobstructed assumption. If the endpoints are closer than that, the direct slant distance is not simply the sum of their individual horizons; it depends on their angular separation around Earth.
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Direct endpoint-to-endpoint geometry when positions are known
Let θ be the central angle between the two endpoint locations, and let their distances from Earth’s centre be r₁ = R + h₁ and r₂ = R + h₂. The straight-line distance between them is:
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S = √(r₁² + r₂² − 2r₁r₂ cos θ)
The line is geometrically clear of the spherical surface when the angular separation does not exceed the sum of the two horizon angles:
θ ≤ arccos(R/(R+h₁)) + arccos(R/(R+h₂))
Use this condition when you have latitude and longitude for both aircraft or when a mission planner needs a visibility boundary rather than only a worst-case range.
Worked example: UAV at 120 m and balloon at 1 km
Assume a spherical Earth with R = 6,371 km. Convert the UAV height to kilometres: 120 m = 0.12 km. The balloon is at h₂ = 1 km.
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D₁ = √(0.12² + 2 × 6,371 × 0.12) ≈ 39.1 km. - Balloon horizon:
D₂ = √(1² + 2 × 6,371 × 1) ≈ 112.9 km. - Maximum ideal geometric separation:
D₁ + D₂ ≈ 152.0 km.
The result means that, in a smooth unobstructed spherical model, the two endpoints can remain above the geometric horizon for roughly this separation. It does not account for hills between the UAV and the tangent point, antenna pattern, atmospheric absorption, camera resolution, or legal operating limits.
FAA radio-horizon approximation
For a first-order radio calculation, FAA engineering guidance gives:
RLOS (nautical miles) = 1.23√h₁ + 1.23√h₂
Enter both heights in feet above mean sea level. The formula treats each endpoint’s radio horizon as 1.23√h nautical miles and adds the two values. It represents an effective Earth radius of approximately four-thirds the true radius, modelling ordinary atmospheric refraction. It is an estimate for radio propagation, not a promise that a particular transmitter and receiver will communicate.
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Radio example: 400-ft UAV and 10,000-ft aircraft
- UAV contribution:
1.23√400 = 1.23 × 20 = 24.6 nautical miles. - Aircraft contribution:
1.23√10,000 = 1.23 × 100 = 123 nautical miles. - Estimated radio horizon:
24.6 + 123 = 147.6 nautical miles.
That is approximately 273 km after multiplying by 1.852 kilometres per nautical mile. The larger value than a vacuum geometric calculation reflects the formula’s refraction model. It still does not include transmitter power, receiver sensitivity, antenna gain and pattern, cable loss, frequency-dependent propagation, interference, or terrain blockage.
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What the radio estimate does not guarantee
FAA guidance describes a transmitted signal becoming tangent to Earth’s radio horizon at a given altitude. Beyond tangency, VHF and higher-frequency signals are not ordinarily received except during anomalous propagation. A link can fail before the calculated horizon because of a ridge, a building, a low antenna, Fresnel-zone obstruction, weak link budget, interference, or a badly oriented antenna. Conversely, unusual atmospheric conditions can occasionally extend reception beyond the normal estimate.
Geometric, visual, and radio line of sight are different tests
| Question | What the calculation answers | What still needs checking |
|---|---|---|
| Can Earth curvature block the ray? | Geometric horizon equations answer this for a spherical Earth. | Local terrain, structures, trees, and the actual Earth shape |
| Can an operator see the aircraft? | Only the physical curvature component | Unaided eyesight, target size, contrast, haze, clouds, lighting, and obstructions |
| Can a radio link work? | A first-order radio-horizon estimate | Terrain profile, antenna heights and patterns, frequency, power, receiver sensitivity, interference, and atmospheric conditions |
| May the flight be conducted? | Nothing about legal permission | Applicable aviation rules, airspace, waivers, and operating limitations |
Altitude rules do not change the horizon equation
For U.S. small-UAS operations, FAA Part 107 guidance generally requires the aircraft to remain within the operator’s unaided visual line of sight and sets a general 400-foot-above-ground limit, subject to stated exceptions and waivers. That 400-foot AGL rule is an operational requirement; it does not mean the geometric horizon is 400 feet away, nor does it turn a radio link into a legal beyond-visual-line-of-sight operation.
When applying the 400-foot value to geometry, distinguish local ground elevation from mean sea level. A UAV at 400 feet AGL over a 2,000-foot mountain is about 2,400 feet MSL, while a balloon reported at 10,000 feet MSL has a different endpoint height above the same reference surface. For regulatory compliance, use the rule and authorization applicable to the flight; for curvature calculations, use consistent physical heights.
Step-by-step calculation workflow
- Define the endpoint and the output. Decide whether you need a straight-line tangent distance, a ground arc, a two-endpoint maximum, or a radio-horizon estimate.
- Choose a reference level. Record each endpoint’s height above the same Earth reference. Convert AGL to MSL when combining it with an aircraft or balloon altitude reported MSL.
- Normalize units. For the NASA geometric equations, use kilometres with a 6,371-km radius or metres with a 6,371,000-m radius. For the FAA radio equation, use feet and nautical miles.
- Calculate one endpoint. Use
√(h² + 2Rh)for the exact result or112.88√hfor kilometres when using the small-height approximation. - Combine two endpoints. Add the two geometric horizon distances for the maximum ideal separation, or use
1.23√h₁ + 1.23√h₂for the FAA radio estimate. - Convert only after calculating. Keep the calculation in its native unit system, then convert kilometres, miles, or nautical miles for presentation.
- Screen the result against reality. Examine terrain and structures, check antenna and camera placement, run a radio link budget where relevant, and verify the applicable aviation rules.
Spreadsheet and calculator forms
With height in kilometres and Earth radius in cell B1, the exact one-endpoint formula is:
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For the common NASA approximation with height in kilometres:
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=112.88*SQRT(B2)
For the surface arc:
=$B$1*ACOS($B$1/($B$1+B2))
For the FAA radio estimate, with heights in feet:
=1.23*SQRT(B2)+1.23*SQRT(B3)
Label every output with its unit and method. A cell containing “152” is ambiguous unless it says whether the value is kilometres of geometric slant range, kilometres of surface arc, or nautical miles of estimated radio horizon.
Common mistakes and how to correct them
Treating altitude as a linear range multiplier
Horizon range follows a square-root law. If altitude increases fourfold, the ideal one-endpoint range doubles, not quadruples. Recalculate rather than scaling a previous result linearly.
Mixing metres, kilometres, feet, and nautical miles
Putting 120 into a kilometres formula when the intended height is 120 metres produces a result 31.6 times too large in the square-root term. Convert first, and keep the FAA formula’s feet-and-nautical-miles convention separate from the geometric calculation.
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For a balloon or aircraft link, the elevated far endpoint contributes its own horizon. Add both endpoint distances for the maximum two-site geometric range.
Calling slant range “ground range”
Distance-measuring equipment that operates on line of sight can report slant range. FAA aeronautical guidance specifically distinguishes that measurement from horizontal distance. Use the surface-arc equation or a suitable horizontal-distance conversion when a map or navigation system requires ground range.
Assuming a clear geometric horizon means a working link
Earth curvature is only the first filter. Terrain can hide the tangent path earlier, and a radio can fail from insufficient link margin or interference. Perform a terrain profile and link-budget analysis after the horizon estimate.
Confusing a legal VLOS requirement with a physics limit
A visual-line-of-sight rule is an operating requirement. It neither changes the Earth-radius calculation nor authorizes a flight merely because a radio link is available.
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How to validate a real UAV-to-balloon or UAV-to-aircraft link
- Map both endpoints. Obtain reliable coordinates, endpoint altitudes, and local ground elevations.
- Build a terrain profile. Check ridgelines and man-made obstructions along the proposed path rather than checking only the two endpoint elevations.
- Use the correct antenna reference. Antenna height above local ground, not merely aircraft altitude, determines whether nearby terrain blocks the first part of the path.
- Run the radio budget. Include frequency, transmit power, antenna gains and patterns, feeder or cable loss, receiver sensitivity, expected interference, and required data rate.
- Apply atmospheric assumptions. Treat the FAA 4/3-Earth radio result as a planning estimate. Do not use anomalous-propagation conditions as a normal design margin.
- Check operations and permissions. Confirm visual-line-of-sight, altitude, airspace, and waiver requirements for the jurisdiction and mission.
- Log the method with the result. Record Earth radius, endpoint heights, units, formula, terrain data, atmospheric model, and date so another operator can reproduce the number.
Choosing the right result for your question
| If you are asking… | Use… | Remember… |
|---|---|---|
| “How far can a drone see before Earth blocks the view?” | One-endpoint geometric horizon | Camera resolution, haze, and target contrast may limit useful visibility first |
| “How far apart can a drone and balloon be?” | Sum of their two geometric horizons | This is the ideal maximum; terrain and endpoint geometry can reduce it |
| “How far might a VHF or higher-frequency link reach?” | FAA two-height radio approximation | Use feet and nautical miles; validate with terrain and a link budget |
| “How many kilometres are between the aircraft and tangent point on a map?” | Surface-arc equation | It is not the same as straight-line slant range |
| “Can I conduct this small-UAS flight?” | Applicable aviation regulations and authorization | Physics calculations do not replace legal compliance |
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