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Radar range equation calculator

The radar range equation gives the farthest range at which a radar still detects a target. The echo falls with the fourth power of range, so doubling the range takes 16 times the power. This calculator works out the maximum range from peak power, antenna gain, frequency, the target’s radar cross section, receiver noise and losses.

Typical values, assumed where not published. The table under the calculator shows which numbers are sourced. A selection of three classes, read 7 November 2026.

Power during the pulse, not the average. Marine radars run 4–25 kW, long-range surveillance radars tens of kW to MW.

Counted twice in a monostatic radar, once on the way out and once on the way back.

Wavelength λ = 3.19 cm.

Bird 0.01 m², person or small aircraft 1 m², cabin cruiser 10 m², airliner 40–100 m² (Skolnik). A car at 77 GHz is usually taken as 10 m².

Minimum detectable signal

The weakest echo the receiver can still pick out of its own noise.

About 1/τ for a matched filter: 2 MHz for a 0.5 µs pulse. With pulse compression, 1 over the full pulse length.

dB

How much noise the receiver adds on top of thermal noise.

dB

13 dB gives a 90 % detection chance at a false-alarm rate of 10⁻⁶ for one pulse (Albersheim).

S_min = −111.0 dBm (kT₀B) + 6 dB + 13 dB = −92.0 dBm

dB

Everything lumped together: feed lines, beam shape, scanning, processing, atmosphere. 6–12 dB is common.

Maximum detection range
12.5
one pulse, steady target, free space
Minimum detectable signal
−92.0 dBm
6.36·10⁻¹³ W · noise floor kT₀B −111.0 dBm
Power for +10 % range
17.6 kW
×1.46 (+1.66 dB) for 13.7 km
Power for 2× range
192 kW
×16 (+12.04 dB) for 25.0 km
Range at 2× power
+18.9 %
14.8 km. At ½ power: −15.9 %, 10.5 km
EIRP
101.8 dBm
peak power + transmit gain

Double the range costs 16× the power: 12 kW → 192 kW to reach 25.0 km.

−120−100−80−60−401101001000range (km, log)dBmS_min = −92.0 dBmP_r ∝ 1/R⁴×16 powerR_max = 12.5 km2·R_max = 25.0 kmecho power vs range · log range, dB power

Echo power at the radar against range, both on log scales. The green line is the echo from your target, falling 40 dB for every tenfold step in range because of the 1/R⁴ law. The amber dashed line is the minimum detectable signal S_min. Where the green line crosses it, at the green dot, is the maximum range R_max. The pale dashed line is the same radar with 16 times the power. It crosses S_min at exactly twice the range, marked with the hollow dot. The range axis spans three decades around R_max.

Single pulse, non-fluctuating target, free space, no clutter, no jamming, no earth curvature. The example radars use typical values, assumed where no figure is published, and are not product specifications. Furuno and TI are trademarks of their owners; Waveshed is not affiliated with them.

Radar terms in one minute

Radar cross section (σ, RCS)
The area of a perfect isotropic reflector that would send back the same echo as the target. It has little to do with physical size. A 30 cm corner reflector can beat a yacht. dBsm is σ in dB relative to 1 m².
Peak power and EIRP
Peak power is what the transmitter puts out during a pulse. EIRP adds the antenna gain. It is the power an isotropic antenna would need to give the same beam.
Minimum detectable signal (S_min)
The weakest echo that still crosses the detection threshold. It is thermal noise kT₀B, raised by the noise figure, plus the signal-to-noise ratio the detector needs.
Matched-filter bandwidth
The receiver bandwidth that passes the pulse and as little noise as possible, about 1/τ for a pulse of length τ. Pulse compression lets a long coded pulse act like a short one while keeping the long pulse’s energy.
Detection probability and false alarms
Raising the threshold cuts false alarms but misses more real echoes. P_d 0.9 at a false-alarm probability of 10⁻⁶ is a common design point and needs about 13 dB SNR on one pulse.
System losses (L)
Everything that eats signal between the transmitter and the detector: feed lines, radome, the beam not pointing straight at the target, scanning, imperfect processing, atmospheric absorption.

Where the example numbers come from

Three classes of radar, not the specifications of any product. Values marked * are assumed, typical figures with no published source. The rest comes from the documents in the sources list.

ExampleP_tGfσBNFSNRLR_max
Marine X-band (FAR-2117 class)12 kW31 dBi9.41 GHz10 m²2 MHz*6 dB*13 dB8 dB*12.5 km
ATC L-band (ARSR-4 class)65 kW36 dBi*1.3 GHz1 m²6.67 kHz3 dB*13 dB10 dB*226 km
Automotive 77 GHz (AWR1843 class)12 dBm10 dBi*77 GHz10 m²*195.3 Hz*14 dB15 dB*3 dB*99.4 m

How it works

  1. 01

    Follow the power out to the target

    The antenna concentrates the transmit power into a beam. At range R the power density is Pt·Gt / (4π·R2), because the power spreads over a sphere of area 4π·R2 and the gain says how much of it goes the right way.

  2. 02

    Let the target send some back

    The target intercepts that density over its radar cross section σ and re-radiates it. On the way back it spreads over another sphere, so the density at the radar is smaller by another 4π·R2. The receive antenna catches it over its effective area Ae = Gr·λ2 / (4π).

  3. 03

    Find the weakest echo the receiver can use

    Thermal noise is −174 dBm per hertz at 290 K. Add 10·log10(B) for the bandwidth, the noise figure and the SNR the detector needs. For 2 MHz, 6 dB and 13 dB that is −111.0 + 6 + 13 = −92.0 dBm.

  4. 04

    Solve for the range and read the R⁴ rule

    Setting the echo equal to Smin gives Rmax as the fourth root of everything else. So every factor works only through its fourth root. Doubling the range takes 16 times the power, gain squared or σ: 12 kW reaches 12.5 km, 192 kW reaches 25.0 km. Doubling the power buys only 19 % more range.

  5. 05

    Know what the single-pulse answer leaves out

    This is one pulse on a steady target in free space. Real radars integrate many pulses and reach farther. Fluctuating targets, rain, sea clutter and jamming pull the range back. The earth’s curve and the terrain can hide a target at any power.

Formulas

Maximum range
Rmax = ( Pt·Gt·Gr·λ2·σ(4π)3·Smin·L )1/4
  • Pt — peak transmit power, W
  • Gt, Gr — transmit and receive gain, linear (Gt = Gr with one antenna)
  • λ — wavelength c/f, m
  • σ — radar cross section of the target, m²
  • Smin — minimum detectable signal, W
  • L — system losses, linear
  • (4π)3 = 1984.4 — the two spheres the power spreads over
  • bistatic: the bracket is (Rt·Rr)2, so the result is Rt·Rr
Minimum detectable signal
Smin (dBm) = −174 + 10·log10(B) + NF + SNRmin
  • −174 dBm/Hz — thermal noise k·T0 at T0 = 290 K
  • B — receiver bandwidth, Hz (≈ 1/τ)
  • NF — receiver noise figure, dB
  • SNRmin — signal-to-noise ratio the detector needs, dB
The R⁴ rule
R2R1 = ( P2P1 )1/4
  • 2× range needs 16× power (+12.04 dB)
  • 2× power gives 1.189× range (+18.9 %)
  • +10 % range needs 1.464× power (+1.66 dB)

Worked example

A 12 kW X-band marine radar and a cabin cruiser (σ = 10 m²)
  1. λ = c / f = 299 792 458 / 9.41·109 = 3.19 cm
  2. Smin = −174 + 10·log10(2·106) + 6 + 13 = −92.0 dBm (6.36·10⁻¹³ W)
  3. G = 31 dBi → 1259, L = 8 dB → 6.31, (4π)3 = 1984
  4. Rmax = ( 12 000 · 12592 · 0.03192 · 10 / (1984 · 6.36·10−13 · 6.31) )1/4 = 12.5 km (6.74 nmi)
  5. → 25.0 km would take 16× the power, 192 kW. 10 % more range (13.7 km) needs 17.6 kW

FAQ

Why does doubling the range need 16 times the power?
Because the signal spreads out twice. It thins as 1/R2 on the way to the target and again as 1/R2 on the way back, so the echo falls as 1/R4. Twice the range means 24 = 16 times weaker, which 12 dB more power, gain squared or σ has to make up. The other way round, twice the power buys only 19 % more range.
What SNR should I enter?
About 13 dB for one pulse, if you want a 90 % chance of detecting a steady target with one false alarm in a million. Albersheim’s approximation gives it: A = ln(0.62/10−6) = 13.34, B = ln(0.9/0.1) = 2.197, SNR = A + 0.12·A·B + 1.7·B = 20.6, which is 13.1 dB. Fluctuating targets need several dB more. Integrating many pulses needs less per pulse.
Why is the published range of my radar longer than this?
Mostly pulse integration. The equation here counts one pulse, while a radar sums the tens of pulses that hit the target in each sweep. The ATC example gives 226 km against a published 250 nmi (463 km). Closing that gap takes about 12.5 dB, which integration supplies. Datasheets also often quote a large target or ideal conditions.
What radar cross section should I use?
Use the target’s σ at your frequency and aspect, and treat it as a rough number. Skolnik’s example values are about 0.01 m² for a bird, 1 m² for a person or a small aircraft, 10 m² for a cabin cruiser and 40–100 m² for an airliner. Automotive radar at 77 GHz usually takes 10 m² for a car. σ swings by 10 dB or more as the target turns, and through the fourth root that moves the range by a factor of 1.78.
Is a bistatic radar different?
Yes. With the transmitter and receiver apart, the equation fixes the product of the two distances, Rt·Rr, not one range. The detection limit is an oval around both sites, and the calculator shows its geometric mean Rt·Rr. Choose bistatic and give the receive antenna its own gain.

The equation says how far your radar could see in empty space. Over real ground, hills and the earth’s curve often decide first. Waveshed draws line-of-sight from your radar site over real elevation data, free in your browser.

Check your radar’s line of sight →

Related guides

  • LOS vs RF analysis — why a strong enough signal still needs a clear path over the terrain
  • Heights: AGL vs AMSL — mast height above ground decides the radar horizon, not altitude on the map

Sources & further reading

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