Radar horizon calculator
The radar horizon is the farthest range at which a radar beam still clears the curved earth to reach a target. With standard refraction it is 4.12·(√h₁ + √h₂) km for the radar and target heights in metres. A 30 m radar sees an aircraft at 10 000 ft out to about 250 km, and nothing below 1 853 m at 200 km.
Example antenna heights. They only fill in the field below.
Above the sea or the ground under the radar, not above sea level on the map.
Example heights. Aircraft altitudes are in feet, as pilots fly them.
Height of the target’s top above the surface beneath it.
k scales the earth radius to stand in for refraction. 4/3 is the standard value.
The radar stands on top of the bulge, so the surface falls away from its foot. The green line is the lowest ray. It leaves the antenna at h₁, grazes the surface at the amber horizon point and keeps dropping. The shaded wedge under it is the radar shadow, where nothing can be seen. The target stands with its top on that ray, which is the farthest it can be seen (first mode) or the lowest it can fly at your range (second mode). Distances are to scale, and the corner note gives the factor the heights are stretched by.
Smooth earth, one k-factor for the whole path, heights above the surface under each end. It says whether the earth’s curve hides the target, not whether the echo is strong enough. That is the radar range equation’s job.
How it works
- 01
Find the radar’s own horizon
The lowest ray leaves the antenna and grazes the surface at d1 = 2·k·R·h1. With R = 6371 km and k = 4/3 that is 4.12·h km for h in metres. A 30 m tower reaches 22.6 km. A boat hull farther out is below the horizon.
- 02
Add the target’s own horizon
A raised target sees over the curve from its side too. Its horizon d2 follows the same formula, and the two meet at the grazing point. An aircraft at 10 000 ft (3 048 m) adds 227.6 km, so the 30 m radar sees it out to 250.1 km (135.1 nmi). Pilots use 1.23·( h1 + h2 ) nmi with heights in feet.
- 03
Turn it round for low flyers
Beyond d1 the grazing ray climbs above the surface by (d − d1)2 / (2·k·R). That is the lowest altitude the radar can see at range d. At 200 km it is 1 853 m (6 079 ft). Anything lower flies in the radar shadow, whatever the transmit power.
- 04
Choose the atmosphere
Radio waves bend slightly downward in normal air, which the 4/3 effective earth radius models. Use k = 1 for pure geometry and 2/3 for subrefraction, which shortens every horizon. Over warm seas, ducting can carry a radar far beyond these numbers, but not reliably.
Formulas
- d — distance to the horizon, km
- k — effective-earth-radius factor (4/3 in a standard atmosphere)
- R — earth radius, 6371 km
- h — height above the surface, m (the 4.12 assumes k = 4/3)
- h1 — radar antenna height, m
- h2 — target height above the surface, m
- at k = 4/3: 4.12·( h1 + h2 ) km with h in m
- aviation form: 1.23·( h1 + h2 ) nmi with h in ft
- d — range to the target, km (hmin = 0 while d ≤ d1)
- d1 — the radar’s own horizon, km
- 2·k·R = 16 989 km at k = 4/3; multiply km by 1000 for m
Worked example
- radar: d1 = 4.12·30 = 22.6 km
- aircraft at 10 000 ft = 3 048 m: d2 = 4.12·3048 = 227.6 km
- d1 + d2 = 250.1 km (135.1 nmi). In feet: 1.23·( 98.4 + 10000 ) = 135.1 nmi
- lowest visible altitude at 200 km: hmin = (200 − 22.6)2 / 16 989 = 1.853 km = 1 853 m (6 079 ft)
- → the radar sees this aircraft out to 250 km, and nothing below 1 853 m at 200 km
FAQ
- Why 4.12·√h and not 3.57·√h?
- 3.57·√h is pure geometry, with k = 1. Normal air bends radio waves slightly downward, as if the earth had 4/3 of its real radius, and that gives 4.12·√h km for h in metres. In feet and nautical miles the same rule is 1.23·√h. The 4/3 value is the median used in radar and link planning.
- Why do low-flying aircraft escape radar?
- Because below the grazing ray, beyond the radar’s horizon, the earth itself is in the way. A 30 m radar sees nothing below about 1 850 m at 200 km. That is why airborne early-warning aircraft and aerostats lift the radar instead of adding power. From 9 000 m the radar’s own horizon is 391 km.
- Does more transmit power push the radar horizon out?
- No. The horizon is geometry and refraction, so power only decides whether an echo inside it is strong enough. Beyond the horizon, diffraction leaks a little energy into the shadow, but far too little for a normal radar. Height is what moves the horizon.
- How far can a ship’s radar see a sea-skimming missile?
- Only about 25 km. A 15 m mast and a 5 m target give 4.12·( 15 + 5 ) = 25.2 km, or 13.6 nmi. At subsonic speed that is under two minutes of warning, which is why warships also rely on helicopters and aircraft for low-altitude cover.
- Is the 4/3 earth always right?
- No. It is a median. Super-refraction and ducting over warm water can carry a radar far beyond the horizon, while subrefraction, with k below 1, shortens it. Use the atmosphere selector for what-ifs, and treat any range beyond the 4/3 horizon as luck rather than coverage.
This is the horizon over a smooth sphere. Around a real radar site, hills and ridges cast their own shadows, often closer in. Waveshed draws line-of-sight from any site over real elevation data, free in your browser.
Map your radar’s real coverage →Related guides
- Heights: AGL vs AMSL — the formula wants height above the surface under each end, not altitude on the map
- Elevation profile & measuring tools — the simulator’s side view shows where a real ridge, not the curve, blocks the beam
Sources & further reading
- M. I. Skolnik, Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001 — chapter 8: propagation of radar waves, refraction and the 4/3 earth
- MIT Lincoln Laboratory — Introduction to Radar Systems (web course) ↗ — lecture 3: propagation effects, video and viewgraphs
- ITU-R Recommendation P.453 — The radio refractive index: its formula and refractivity data ↗ — refractivity gradients, super-refraction and ducting
- ITU-R Recommendation P.530 — Propagation data for terrestrial line-of-sight systems ↗ — the effective-earth-radius factor k used in planning
- Radar horizon — Wikipedia ↗ — definition, geometric form and the radar shadow (read 7 November 2026)
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