Radio horizon calculator
The radio horizon is the farthest point a signal reaches over smooth earth before the surface itself blocks the path. It grows with the square root of antenna height, about 4.12·√h km for a height h in meters once standard atmospheric refraction is included. Two stations each contribute their own horizon, so the maximum line-of-sight distance is the sum.
Height above local ground, not above sea level.
k scales the earth radius to stand in for refraction.
The green line is the sight ray from mast A. It just grazes the surface at the amber grazing point, then carries on to mast B. Each antenna height buys one of the two distances marked along the ground, d₁ and d₂, and the link reaches as far as their sum. Heights are exaggerated. The two distances are drawn in proportion.
Smooth-earth model with a single k-factor for the whole path. Real terrain almost always sets a nearer limit, which is exactly what a viewshed simulation computes.
How it works
- 01
Start with the geometry
Over a smooth sphere, the ray leaving an antenna at height h grazes the surface at d = 2·R·h. With R = 6371 km that is 3.57·h km for h in meters: 19.6 km from a 30 m tower, 5.0 km from a 2 m handheld.
- 02
Add refraction with the k-factor
Air density falls with altitude, so rays bend gently downward and follow the curve farther than pure geometry allows. The standard trick is to keep the rays straight and inflate the earth radius to k·R instead. The median mid-latitude value is k = 4/3, which stretches every horizon by 4/3, or 15%.
- 03
Add the two horizons
Each station has its own horizon, and the link closes while both rays reach the same grazing point: d = 3.57·( k·h1 + k·h2 ), which becomes 4.12·( h1 + h2 ) at k = 4/3. Returns diminish fast. Doubling a height adds only 41% to that station’s horizon, worth 2.4 km on a 2 m handheld and 9.3 km on a 30 m tower.
- 04
Treat the result as a ceiling
This is the smooth-earth best case. Hills, buildings and tree lines between the stations cut it short, and the first ridge that pokes above the sight line ends the link long before the horizon does. Signals do arrive beyond the horizon by diffraction, troposcatter and ducting, but tens of dB weaker and not on a schedule.
Formulas
- d — distance to the radio horizon, km
- k — effective-earth-radius factor (4/3 in a standard atmosphere)
- R — earth radius, 6371 km
- h — antenna height above ground, m
- h1, h2 — the two antenna heights, m
- dtotal — maximum line-of-sight distance, km
- at k = 4/3 this becomes dtotal = 4.12·( h1 + h2 )
Worked example
- tower: d1 = 4.12·30 = 22.6 km
- handheld: d2 = 4.12·2 = 5.8 km
- dtotal = d1 + d2 = 28.4 km
- without refraction (k = 1): 3.57·( 30 + 2 ) = 24.6 km
- → beyond ~28 km even perfectly flat ground blocks this link
FAQ
- Why is the radio horizon farther than the visible horizon?
- Both light and radio bend slightly downward in the atmosphere, but radio bends more under standard conditions. The geometric horizon is 3.57·h km, the visible horizon about 3.86·h, and the radio horizon 4.12·h, roughly 15% beyond pure geometry.
- Does frequency change the radio horizon?
- Not the horizon distance itself. It follows from geometry and refraction, and both are frequency-independent at radio wavelengths. What happens beyond the horizon is very frequency-dependent. Low VHF diffracts into the shadow reasonably well, while at 5 GHz the shadow behind the bulge is deep and starts abruptly.
- Can a signal reach beyond the radio horizon?
- Yes, but not dependably. Diffraction leaks some energy into the shadow, tropospheric scatter supports 100 km-plus paths with high power and large antennas, and ducting under a temperature inversion can carry a signal for hundreds of kilometers. None of these are something to plan a link around.
- What does k = 4/3 actually assume?
- A standard atmosphere: a refractivity gradient near −40 N-units per kilometer through the lowest layer, typical of median mid-latitude conditions. Real gradients follow temperature and humidity. Subrefractive weather pulls k toward 2/3 and shortens the horizon, while strong inversions over water push it past 4/3 and into ducting.
This is your horizon over a billiard-ball earth. The real one is made of hills. Waveshed draws line-of-sight over actual elevation data, free in your browser.
Compute your real horizon →Related guides
- Elevation profile & measuring tools — the simulator’s side view shows where the real terrain, not the horizon, ends the path
- Heights: AGL vs AMSL — the formula wants height above ground, not above sea level, and that mix-up ruins most horizon estimates
Sources & further reading
- ITU-R Recommendation P.530 — Propagation data for terrestrial line-of-sight systems ↗ — source of the effective-earth-radius k-factor used in link planning
- Line-of-sight propagation — Wikipedia ↗ — radio horizon, refraction and the 4.12·√h form
- Horizon — Wikipedia ↗ — geometric derivation of √(2·R·h) and the refracted variants
Related tools
Earth bulge calculator
Height the Earth’s curvature adds mid-path, with k-factor refraction.
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First Fresnel zone radius and the 60% clearance rule for link paths.
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Distance to the horizon from any height, and how much of a distant peak or ship the curve hides.
Walkie-talkie range calculator
How far two handhelds really reach (PMR446, FRS, GMRS, ham HTs) from height, power and terrain type.