Radar Horizon Calculator
Enter your radar antenna height and target height to find the radar horizon distance, the target horizon distance, and the maximum range at which the radar can detect the target. The calculator uses the standard 4/3 Earth-radius model for atmospheric refraction - the same model used in aviation and maritime radar planning. Switch between metric, imperial, and nautical units instantly, and toggle between the refraction model and pure geometry to see the difference.
What is the radar horizon?
The radar horizon is the maximum distance at which a radar antenna can detect a target at the same height as the antenna. Because Earth is curved, the line of sight from any elevated point follows the planet's surface and eventually dips below it. A target beyond that geometric limit is simply hidden by the curvature of the Earth, no matter how powerful the radar is. For a radar antenna sitting 30 m above sea level, the radar horizon is roughly 7.5 nautical miles; raising the antenna to 100 m extends it to about 13.8 nm. The same physics applies to visual lookouts, optical rangefinders, and radio links - only the exact correction factor differs.
How atmospheric refraction extends radar range
In practice, radar waves do not travel in perfectly straight lines. The atmosphere is denser at lower altitudes, which bends radio waves slightly downward - an effect called atmospheric refraction. Under standard atmospheric conditions, this bending is equivalent to making the Earth appear about 4/3 times its actual radius for radar propagation purposes. This k = 4/3 factor is the basis of the standard radar horizon formula: d = sqrt(2 times (4/3) times R_E times h). The refraction typically adds around 15-16% to the purely geometric horizon, which matters a great deal in maritime radar planning. Under unusual conditions - such as temperature inversions over warm seas - superrefraction (k up to 2 or more) can dramatically extend radar range in what is called ducting, while subrefraction compresses it.
Radar horizon vs. maximum detection range
The radar horizon of the antenna itself tells you how far the radar can see a target sitting on the sea or ground. But when the target is elevated - an aircraft at 3,000 m, or a ship with a tall mast - it has its own horizon measured from its own height. The total maximum detection range is the sum of both horizons: D = d_radar + d_target. A radar at 30 m height and an aircraft at 3,000 m altitude, for example, have a combined geometric visibility of roughly 170 km under standard refraction. This two-horizon model is why aircraft are visible on radar at ranges far beyond the ground-level radar horizon - and why airborne early-warning (AEW) aircraft place their radars very high to maximise the antenna horizon.
Practical limits beyond the horizon formula
The horizon formula gives the geometric detection limit, but the actual usable range of a radar system depends on several additional factors. Transmit power and antenna gain determine how strong the reflected signal is at range. Target radar cross-section (RCS) - the effective reflective area of the target - varies enormously: a large cargo ship may have an RCS of tens of thousands of square metres while a small fibreglass boat may be nearly invisible. Sea clutter - returns from the wave tops - masks small targets near the antenna in rough conditions. Rain and fog attenuate the signal, especially at higher radar frequencies. Finally, processing algorithms (constant false-alarm rate, CFAR) filter clutter to reveal real targets. For all these reasons, the horizon distance is a best-case ceiling, not a guaranteed detection range.
Radar horizon by antenna height (standard atmosphere, k = 4/3)
| Antenna height | Radar horizon (nm) | Max range for 5 m target (nm) | Max range for 50 m target (nm) |
|---|---|---|---|
| 3 m (10 ft) | 2.4 | 5.7 | 11.4 |
| 5 m (16 ft) | 3.1 | 6.4 | 12.1 |
| 10 m (33 ft) | 4.4 | 7.7 | 13.4 |
| 20 m (66 ft) | 6.2 | 9.5 | 15.2 |
| 30 m (98 ft) | 7.6 | 10.9 | 16.6 |
| 50 m (164 ft) | 9.8 | 13.1 | 18.8 |
| 100 m (328 ft) | 13.8 | 17.1 | 22.8 |
| 200 m (656 ft) | 19.5 | 22.8 | 28.5 |
| 500 m (1640 ft) | 30.8 | 34.1 | 39.8 |
| 1,000 m (3281 ft) | 43.6 | 46.9 | 52.6 |
Approximate radar horizon and maximum detection range for a target at sea level. Values in nautical miles. Computed using d = sqrt(2 x (4/3) x R_E x h).
Frequently asked questions
What is the formula for the radar horizon?
The standard formula is d = sqrt(2 times k times R_E times h), where d is the horizon distance, k is the refraction factor (4/3 under standard atmospheric conditions), R_E is the Earth radius (6,371 km), and h is the antenna height in the same length unit as R_E. For practical use with heights in metres and distances in nautical miles, the approximation is d (nm) approximately 2.23 times sqrt(h in metres). Note that the exact coefficient depends on the refraction k-factor you apply.
Why is the k-factor 4/3?
Radio waves in the atmosphere are refracted by the vertical gradient of air density. Under the standard atmospheric model (ISA), this bending is strong enough to make electromagnetic waves behave as if the Earth were 4/3 times its actual radius. Using k = 4/3 in the horizon formula accounts for this effect without needing to model the full ray path through the atmosphere. In reality k varies with weather: warm humid air over cool water (common in tropics) can push k to 2 or higher, creating radar ducting where signals travel much farther than normal.
How do I calculate the maximum range for an airborne target?
Add the radar horizon of the antenna to the horizon of the target. For a radar at height h_r and a target at height h_t, the maximum range is D = sqrt(2 times k times R_E times h_r) + sqrt(2 times k times R_E times h_t). Enter both heights in this calculator: the Radar horizon field gives the antenna contribution, the Target horizon gives the target contribution, and the Maximum detection range is their sum.
What is the difference between radar horizon and visual horizon?
The visual (optical) horizon uses essentially the same spherical geometry, but light is refracted less than radio waves under standard conditions. The optical k-factor is approximately 1.07-1.13, versus 4/3 (approximately 1.33) for radar. This means the radar horizon extends a little farther than the visual horizon - roughly 6% farther under standard conditions. In practice, a human lookout, a ship's radar, and a laser rangefinder will all have slightly different horizons because of these different k-factors.
Does terrain or sea state affect the radar horizon?
The formula assumes a smooth sphere with a uniform atmosphere. Terrain adds effective height to the antenna if it is on a hill, but it also creates shadows (radar shadow zones) behind high ground. At sea, waves can raise small targets slightly above sea level, which marginally increases their horizon. Sea clutter from wave tops often limits effective detection of small targets to well below the geometric horizon because the clutter return can mask the target echo even though it is in line of sight.
Why does raising the antenna height improve radar range so much?
Because the horizon formula involves the square root of height, doubling the antenna height multiplies the radar horizon by sqrt(2) approximately 1.41, a 41% increase. Quadrupling the height doubles the horizon. This square-root relationship means that early gains in height are large: going from 5 m to 10 m adds roughly 1.3 nm, while going from 100 m to 105 m adds only about 0.3 nm. This is why maritime radars are placed as high as practical on a mast, and why AEW aircraft achieve dramatically greater coverage than land-based radars of comparable power.
What is radar ducting?
Radar ducting (superrefraction) occurs when the atmosphere has an unusually strong refractivity gradient, often caused by warm dry air over cool moist sea air (an evaporation duct) or a temperature inversion. In duct conditions the k-factor can reach 2 to 4 or more, and radar signals can travel far beyond the normal horizon, sometimes wrapping around the Earth's curvature for hundreds of kilometres. Ducting can extend detection dramatically but also causes false echoes from distant clutter. Use the Superrefraction (k = 2) option in this calculator for an estimate under duct conditions.