A radar horizon calculator answers a geometric question with an atmospheric correction bolted on. Radio waves travel almost in straight lines, so an antenna cannot see past the curve of the earth. But the atmosphere bends them slightly downward, so the practical horizon sits further away than pure geometry predicts. The standard way to handle that is the 4/3 effective earth radius, and this page states that assumption explicitly because it is doing a lot of work.
Arb Digital publishes free physics calculators that name their assumptions rather than burying them. The earth curvature calculator covers visual horizon distance, surface drop and hidden height and offers several refraction settings including the geometric case. This page is the radio version: it adds the two horizons together, defaults to the 4/3 radio convention, and shows the geometric answer alongside so the size of the correction is visible.
What This Radar Horizon Calculator Does
It computes the distance from each end of the link to its own horizon, then adds them. A radar at 30 metres can see its own horizon; a target at 100 metres can be seen from its horizon; the two meet, and the sum is the maximum range at which any line of sight exists at all.
Everything is calculated on an effective earth of radius k times the true radius. With k = 4/3 the rays can be drawn as straight lines over a fictitiously larger earth and the geometry comes out right for a standard atmosphere. Setting k to 1 removes the correction entirely and gives the true geometric horizon, which the calculator also reports as a comparison.
Results are given in kilometres, nautical miles and statute miles, because radar, marine and aviation practice use different ones. The two component horizons are shown separately so you can see which end of the link is limiting.
How to Use It
- Enter the antenna height above the surface. Mast height plus site elevation for a land radar, height above waterline for a ship.
- Enter the target height. Set it to zero for a surface target and the range collapses to the antenna's own horizon alone.
- Choose the k factor. Leave it at 4/3 for standard conditions, or use the subrefractive and superrefractive options to see how far the answer can move.
- Compare against the geometric figure. The gap between the two is entirely the refraction assumption, and it is around 15 per cent.
- Treat the answer as a ceiling, not a detection range. Line of sight is necessary for detection, not sufficient for it.
The Formula: How the Radar Horizon Is Calculated
For a height h that is small compared with the earth's radius, the distance to the horizon is d = √(2Reh), where Re is the effective earth radius kR. The line-of-sight transmission section of Engineering LibreTexts derives this from the exact expression √(2hR + h²), showing that the h² term is negligible because antenna heights are tiny compared with the earth's radius. The total link range is the sum of the two individual horizons.
The refraction correction itself is the standard treatment of tropospheric bending. ITU-R Recommendation P.834, on the effects of tropospheric refraction on radiowave propagation, is the international reference for how the atmosphere's refractive index gradient bends radio rays and how that is accounted for in propagation work. The 4/3 factor is the convenient shorthand for a standard vertical gradient of refractivity: it lets you draw the ray as a straight line on an inflated earth instead of tracing a curved path over the real one.
Work the defaults by hand. With k = 4/3 and R = 6,371 km, the effective radius is 8,494.7 km. The antenna at 30 m gives √(2 × 8,494,667 × 30) = √509,680,000 = 22,576 m, or 22.58 km. The target at 100 m gives √(2 × 8,494,667 × 100) = 41,218 m, or 41.22 km. The sum is 63.79 km, which is 34.45 nautical miles or 39.64 statute miles. Set k to 1 and the same heights give 19.55 km and 35.70 km for a total of 55.25 km — about 15.5 per cent less.
The familiar shortcut d(km) ≈ 4.12√h(m) is exactly this formula with k = 4/3 folded into the constant. Check it: 4.12 × √30 = 4.12 × 5.477 = 22.57 km, matching the long calculation.
What the 4/3 Assumption Really Means
Air is denser near the surface, so its refractive index falls with height and a radio ray curves gently downward. Rather than integrate that curvature, propagation engineers replace the real earth with a larger fictitious one on which the ray travels straight. If the refractivity gradient has its standard value, that fictitious radius works out at four thirds of the real one.
The number is therefore a stand-in for one particular atmospheric condition. It is a good average and it is the right default, but it is an average of something that varies continuously through the day, with season, with latitude and with weather. Treating 4/3 as a constant of nature is the most common misuse of this calculation.
Subrefraction happens when the gradient is weaker than standard, which occurs when a cold surface chills the air above it or when humidity rises with height. The effective radius drops, k can fall to 0.7 or lower, and the horizon closes in. In hilly terrain that can turn a marginal link into a blocked one.
Ducting: When the Horizon Stops Applying at All
Superrefraction is the opposite case, and in its extreme form it does not just extend the horizon — it removes it. When a strong temperature inversion or a sharp drop in humidity with height creates a steep enough refractivity gradient, the ray curves more sharply than the earth does and becomes trapped in a layer. That layer acts as a waveguide, and signals travel far beyond any horizon calculation.
Evaporation ducts form routinely over warm water, typically a few tens of metres deep, and are a normal feature of maritime radar operation. Surface-based and elevated ducts form under high-pressure subsidence inversions. Under ducting conditions a marine radar may paint targets hundreds of kilometres away, and terrestrial links that normally do not interfere with each other suddenly do.
None of this is captured by a k factor. Ducting is not a larger effective earth; it is a different propagation mechanism. A calculator like this one describes the standard case and tells you where the horizon would be. It cannot tell you when the atmosphere has stopped behaving that way, which is why operational radar work uses measured or forecast refractivity profiles rather than a fixed assumption.
Line of Sight Is Not Detection Range
The horizon sets an absolute ceiling. Nothing below it can be seen, and the radar equation governs everything at shorter ranges. Whether a target inside the horizon is actually detected depends on transmitted power, antenna gain, wavelength, target radar cross-section, receiver noise and the integration the processor performs — a chain the free space path loss calculator and the EIRP calculator address on the link-budget side.
There is also a geometric subtlety the horizon formula hides. A path can be clear in the sense that the straight line misses the earth, and still be obstructed in practice, because a radio beam is not a line. It occupies a volume, and if terrain intrudes into the first Fresnel zone around the path there is diffraction loss even with nominal line of sight. The Fresnel zone calculator handles that clearance requirement, and for point-to-point links it usually bites before the horizon does.
Finally, terrain. This calculation assumes a smooth spherical surface. A hill halfway along the path blocks the link regardless of what the horizon arithmetic says, and a valley floor site can have a horizon far closer than its height above sea level suggests. Antenna work on the transmit side is covered by the dipole antenna calculator.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using height above local ground instead of above the surface — what matters is height above the reflecting surface, which for a hilltop site includes the hill.
- Forgetting the target's own horizon — the range is the sum of two horizons, and for a high-flying aircraft the target term dominates completely.
- Treating 4/3 as a physical constant — it represents one standard atmospheric condition, and real gradients vary continuously.
- Confusing radio horizon with visual horizon — optical refraction is weaker, so the visual horizon is closer than the radio one for the same eye height.
- Reading the result as a detection range — it is a geometric ceiling. Power, gain, target size and receiver sensitivity decide what is actually seen inside it.
Related Free Tools From Arb Digital
The earth curvature calculator covers visual horizon, surface drop and hidden height with selectable refraction. For link budgets use the free space path loss calculator and the EIRP calculator, and for path clearance the Fresnel zone calculator. Antenna dimensions come from the dipole antenna calculator, wave geometry from the wavelength calculator, and unit conversions from the speed converter when you are working timing rather than distance. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
It is a way of accounting for atmospheric refraction by replacing the real earth with a fictitious one four thirds as large, on which radio rays can be drawn as straight lines. It corresponds to a standard vertical gradient of atmospheric refractivity.
The geometric horizon uses the true earth radius and ignores refraction entirely. The radar horizon applies the effective radius, which for standard conditions extends the range by roughly 15 per cent because the atmosphere bends rays gently downward.
Only approximately, and only under standard conditions. The refractivity gradient changes with time of day, season, humidity and weather, so the effective k factor moves. Subrefraction closes the horizon in, and ducting can extend detection far beyond any horizon figure.
Ducting occurs when a strong temperature inversion or humidity gradient traps radio energy in a layer that behaves like a waveguide. Signals then travel far past the calculated horizon. It is a different propagation mechanism, not a larger effective earth, so no k factor represents it.
Because line of sight exists as far as the point where the antenna's horizon and the target's horizon meet. Each end contributes its own distance, which is why raising either the radar or the target extends the range.
No. It is the geometric ceiling: nothing below it can be seen at all. Whether something inside it is detected depends on transmitted power, antenna gain, target radar cross-section and receiver sensitivity, which the radar equation handles separately.
It is the same square-root formula with the 4/3 effective radius folded into a single constant, giving distance in kilometres from height in metres. For 30 metres it gives 22.6 kilometres, matching the full calculation.
This tool is provided for educational and study use. It assumes a smooth spherical earth and a single effective-radius factor representing one atmospheric condition. Real propagation varies continuously with refractivity, and ducting, subrefraction, terrain obstruction and Fresnel zone intrusion are not modelled. It is not a substitute for a measured refractivity profile or an approved propagation-planning tool.
