How this instrument works
Radar horizon distance is the farthest range at which a radar's line of sight to a target at the same altitude is not yet blocked by the curve of the Earth. Height is what buys that range: lift the antenna and the tangent line from its aperture reaches farther before the ground curls away underneath it. The formula behind this page, d = 1.23·√h, converts antenna height in feet directly into that reach in miles — a 100 ft mast reaches 12.3 miles, a 400 ft tower reaches 24.6 miles, four times the height for only twice the range, because the relationship runs on a square root, not a straight line.
The constant 1.23 is not the bare geometric answer; pure line-of-sight trigonometry on a smooth sphere with no atmosphere gives a smaller coefficient, closer to 1.06. The extra reach comes from atmospheric refraction: air near the ground is slightly denser than air higher up, so a radio wave curves gently downward as it travels rather than shooting off in a dead straight line. Engineers fold that bending into the geometry itself by treating the Earth as if it had 4/3 its true radius — the standard '4/3-Earth' model — which is why the radar horizon reaches meaningfully past what a ruler and a globe alone would predict, and why the equivalent formula for visible light, which the atmosphere bends less than radio, uses a smaller coefficient again, nearer 1.17.
This number is a geometric ceiling, not a promise about detection. A radar rated for a 200-mile range against a large aircraft can still miss a target flying below its horizon at 30 miles, because no transmitter power or receiver sensitivity can bend a straight beam around a curved planet — that separate limit belongs to the radar range equation, which involves power, antenna gain, and target reflectivity. The 4/3 figure also assumes a 'standard atmosphere'; a temperature inversion over warm water can duct a beam far beyond this line, while an unusually cold, dense surface layer can pull it noticeably shorter, which is why real coverage diagrams get checked against local weather, not height alone.
- Enter the radar's mounting height into 'Antenna height above ground/sea level, ft' — measured from the antenna itself, not the base of its tower.
- Read 'Radar horizon distance, miles' — the farthest range at which the beam can still reach a target sitting at that same altitude.
- For an elevated target such as an aircraft or a ship's mast, run the calculator a second time with the target's own height and add the two horizon distances together.
- Treat the result as a ceiling set by standard atmospheric refraction; ducting or a temperature inversion can shift the true value in either direction.
Worked example — a 100 ft radar tower's horizon
Set 'Antenna height above ground/sea level, ft' to 100, a realistic mounting height for a coastal surveillance or airport radar tower. The formula gives d = 1.23 × √100 = 1.23 × 10 = 12.3 miles, exactly what 'Radar horizon distance, miles' returns. Inside that 12.3-mile circle, a target flying at the same 100 ft altitude sits within the radar's geometric line of sight; a little beyond it, the curve of the Earth begins to hide even a strong reflector no matter how much power the transmitter pushes.
Raise the same tower to 400 ft and the horizon does not grow fourfold — it grows to 1.23 × √400 = 1.23 × 20 = 24.6 miles, exactly double. That square-root relationship explains why radar towers keep getting taller for diminishing gains: quadrupling height only doubles reach, so a site chasing another 50 percent of range needs roughly 2.25 times its current antenna height, not 1.5 times it.
Questions
Can a powerful radar detect a target flying below its calculated horizon?
No, not from geometry alone. The radar horizon is a straight-line-of-sight limit set purely by antenna height and Earth's curvature; a target flying below that line is physically hidden behind the planet, and no increase in transmitter power, antenna gain, or receiver sensitivity restores a sight line that geometry has already blocked. This is exactly why low, sea-skimming aircraft are so hard to detect from a shore-based or shipboard radar until they close well inside this range.
Why is the constant 1.23 rather than the roughly 1.06 that plain geometry gives?
Because 1.23 already includes atmospheric refraction. Pure trigonometry on a bare, airless sphere gives a coefficient near 1.06; radio waves, however, bend gently downward as they cross the real atmosphere's density gradient, letting the beam follow a little more of the planet's curvature than geometry alone allows. Engineers model that bending by treating the Earth as if its radius were 4/3 the true value, which raises the coefficient from about 1.06 to 1.23, roughly a 16 percent gain in reach.
How is this different from the visual horizon I can see from a hilltop?
The physics has the same shape, but the bending is not identical. Visible light also refracts through the atmosphere, but by less than radio waves do under standard conditions, so optical horizon formulas typically use a smaller effective-Earth-radius factor, close to 7/6 rather than 4/3, giving a coefficient nearer 1.17 instead of 1.23. A radar can therefore usually reach a little farther along the curve than your eyes can from the same height.
Does this account for the height of the target itself, like an aircraft or a ship?
No, this formula only accounts for the antenna's own height above the surface. For a target that also sits above ground — an aircraft at altitude, a ship's mast, a communication tower — run the same 1.23·√h formula a second time using the target's height, then add the two horizon distances together. That combined figure is the range at which the radar's sight line first reaches the target's own sight line, the standard two-point radio line-of-sight range.
What happens at zero antenna height, and does the formula still make sense there?
It returns zero miles, which is the correct idealized limit: an antenna sitting exactly at ground or sea level has no elevated vantage point, so its geometric line of sight along a curved surface vanishes. Real installations never sit at that literal limit — even a handheld unit or a low mast carries a few feet of height — but the zero case is a useful check that the formula behaves the way the underlying geometry demands.
Can real-world conditions make the actual radar horizon different from this figure?
Yes. The 1.23 constant assumes a 'standard atmosphere' with a typical, steady rate of temperature and humidity change with height. A temperature inversion, common over warm water under cooler air, can duct radio waves and extend real range past this figure by tens of miles. A colder, denser surface layer can do the opposite and pull the true horizon in closer, which is why permanent radar sites get checked against local atmospheric statistics rather than the standard constant alone.