Radar Horizon Calculator - Antenna-to-Target Line of Sight

A radar horizon calculator finds the maximum slant range at which a radar can detect a target, given the antenna height, the target height, and whether the 4/3 atmospheric refraction correction is applied. The radar horizon is the distance from the radar to its own line of sight along the curved Earth; the target visibility is the matching distance on the target's side; the sum is the total detection distance. Use this calculator when sizing an air surveillance radar, a coastal surface search radar, or an AWACS mission.

• • Air surveillance radar planning: Estimate how far a primary or secondary surveillance radar can see a small target before the Earth curves below the line of sight.
• • Coastal and shipboard radar geometry: Compute the horizon distance for a 10 m to 25 m mast against a periscope or small boat.
• • AWACS and high-altitude early warning: Check that an airborne radar at 9,150 m can still see a low-flying target before weapon release range.

The result is a planning number, not a hard detection range. Real radars also depend on transmitter power, antenna gain, target radar cross section, and the clutter zone near the ground. The calculator isolates the line-of-sight geometry so the rest of those numbers can be added on top.

For a visual horizon check rather than a radio one, the Binoculars Range Calculator applies the same Earth-curvature geometry to the height of an observer and a distant object.

The calculator applies the Pythagorean theorem to a sphere of radius R_E and sums the two horizon distances. With refraction on, the 4/3 effective Earth radius correction is applied to both square roots.

• k (refraction factor): 1 for the pure geometric horizon, 4/3 for the standard atmospheric refraction correction used by radar textbooks and ITU-R P.528.
• R_E (Earth radius): Mean Earth radius of 6371.009 km from the IUGG Geodetic Reference System 1980, adopted by NIST and used in propagation models.
• h_r (radar antenna height): Height of the transmitter and receiver above local sea level in metres, converted to kilometres before the formula is applied.
• h_t (target height): Height of the target above local sea level in metres, also converted to kilometres before the formula is applied.

When refraction is off, k is set to 1 and the formula returns the pure geometric horizon. For a 10 m radar against a 122 m target the pure geometric distances fall to 11.29 km and 39.42 km, dropping the maximum detection distance to 50.71 km.

According to ITU-R Recommendation P.528-5 propagation curves, the standard 4/3 effective Earth radius correction is the convention used for VHF, UHF, and SHF propagation curves because it approximates the bending of radio waves in a standard atmosphere without requiring a full ray-trace through a refractivity profile.

The 4/3 effective Earth radius correction is a stand-in for the real temperature-versus-altitude profile that drives atmospheric refraction, and the Altitude Temperature Calculator in education and academic returns the standard atmosphere values behind that correction.

Four ideas appear in every radar horizon discussion and on the calculator outputs: geometric horizon, radar horizon, 4/3 effective Earth radius, clutter zone.

Doubling the antenna height multiplies the radar horizon by sqrt(2), about a 41 percent increase, while doubling the transmitter power only changes the signal-to-noise ratio by 3 dB.

The bending of a radio wave through a stratified atmosphere is the same Snell-style refraction problem the Angle of Refraction Calculator solves for an optical interface, only applied to many thin atmospheric layers instead of a single surface.

Set the refraction toggle first and then the two heights. The result rows update as you type.

1. 1 Pick the refraction mode: Select Yes to apply the 4/3 effective Earth radius correction used in operational radar. Select No for the pure geometric horizon.
2. 2 Enter the radar antenna height: Type the antenna height above local sea level in metres. Use 10 m for a shipboard mast, 25 m for coastal radar, and 9,150 m for an AWACS aircraft.
3. 3 Enter the target height: Type the target height above local sea level in metres. Use 122 m for a low-flying aircraft, 3 m for a small boat, 0 m for sea level.
4. 4 Read the radar horizon: The first row shows the radar horizon - the distance from the radar to its own line of sight along the curved Earth.
5. 5 Read the target visibility: The second row shows how far the target can sit from the radar before the Earth's curvature hides it.
6. 6 Add the two for the maximum detection distance: The third row sums the radar horizon and the target visibility. This is the maximum slant range the radar geometry alone allows.

Suppose you are planning a coastal radar at 25 m against a 5 m small boat. Set refraction to Yes, radar height to 25, and target height to 5. The radar horizon reads about 20.6 km and the target visibility about 9.2 km, so the maximum detection distance is about 29.8 km. Raise the mast from 25 m to 40 m and the horizon rises to about 26 km.

If you want to sanity-check the antenna height by measuring it from the ground with a known baseline, the Tree Height Calculator in education and academic turns a distance plus an angle into the same vertical height value used in the radar horizon formula.

The radar horizon calculator turns a textbook formula into three numbers you can paste into a planning slide.

• • Two-mode refraction toggle: Switch between the 4/3 effective Earth radius used in operational radar and the pure geometric value used in optics without retyping the heights.
• • Three outputs in one view: Radar horizon, target visibility, and total maximum detection distance are shown side by side so a planning table can copy them directly.
• • Worked AWACS and ground-radar examples: Default heights match published reference examples, so the page acts as a cross-check against existing calculators.
• • Plain-language scenario row: The scenario row translates the radar antenna height into a deployment context such as AWACS, hilltop radar, or coastal mast.
• • Guarded against zero heights: Zero radar height returns a zero horizon and a maximum detection distance equal to the target visibility; zero target height returns a zero target visibility.

The biggest planning lever is the radar antenna height, not the transmitter power. Doubling the mast height increases the radar horizon by about 41 percent (sqrt(2)), while doubling the transmitter power only changes the link budget by 3 dB.

The radar horizon is a geometric ceiling, and the Harmonic Wave Equation Calculator shows how the amplitude, wavelength, and propagation speed of the underlying electromagnetic wave combine so you can sanity-check the radar band you are planning for.

Four factors change the result even when the heights and the refraction toggle look identical.

• • The formula uses the small-height approximation sqrt(2 R_E h), with an error below 1 percent for heights under 250 km. For very high platforms, fall back to the full Pythagorean form sqrt((R_E + h)^2 - R_E^2).
• • The 4/3 effective Earth radius correction assumes a standard atmosphere. A surface duct extends the horizon further, and a low-lying cold layer shortens it.
• • The calculator does not include transmitter power, antenna gain, receiver noise figure, or target radar cross section. Use the radar horizon as a ceiling rather than an upper bound.

For high-altitude platforms above 250 km the small-height approximation starts to drift, so satellite altimetry uses the full Pythagorean expression. The calculator sticks with the simplified form because radar textbooks and ITU-R P.528 quote that version for terrestrial planning.

According to the NASA Planetary Fact Sheet, the mean radius of Earth is 6371 km, the IUGG GRS80 geodetic value adopted by ITU-R propagation models and used in the radar horizon formula as the R_E constant.

When the radar horizon is short, you often need to confirm the slant range against the actual horizontal separation between the radar site and the target, and the Distance, Midpoint & Slope Calculator in education and academic returns that ground distance plus the line-of-sight elevation angle in one calculation.