Calculate the maximum range at which a monostatic radar can detect a target using the radar range equation. Adjust the transmit power, antenna gain, frequency and target RCS to see the maximum detection range, received echo power and radar band update in real time, and feel how radio power fades with the fourth power of range.
Parameters
Transmit power P_t
W
Peak power the radar delivers to the antenna
Antenna gain G
dBi
How tightly the antenna focuses energy (shared for Tx and Rx)
Frequency f
GHz
Transmit frequency. Sets the wavelength λ and the radar band
Target RCS σ
m²
Radar cross section — the equivalent area of reflection strength
Min. detectable signal S_min
dBm
Lowest power the receiver can identify as a signal
System loss L
dB
Combined loss from feed lines, atmosphere and signal processing
Results
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Max detection range R_max (km)
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Wavelength λ (cm)
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Radar band
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Echo power @ R_max (dBm)
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Antenna gain (linear) G
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Detection class
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Radar detection — pulse and echo animation
A transmit pulse spreads from the radar on the left, reaches the target, and a far weaker echo returns. The fading wavefront intensity shows the 1/R⁴ power falloff.
Maximum detection range R_max [m] of a monostatic radar. P_t: transmit power, G: antenna gain (linear), λ: wavelength, σ: RCS, S_min: minimum detectable signal, L: system loss.
Received echo power P_r [W] at range R. The echo power falls as 1/R⁴ — once on the way out, once on the way back.
What is the Radar Range Equation?
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A radar sends out radio waves to find aircraft, right? What actually decides how far it can see?
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The thing that wraps all of that into a single formula is the "radar range equation". Roughly speaking, the maximum detection range is set by four things: how much transmit power the radar can put out, how tightly the antenna can focus the beam (its gain), how strongly the target bounces the wave back (its RCS), and how faint a signal the receiver can still pick up. Move the sliders on the left and you'll see R_max change in real time.
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More transmit power should mean seeing further — but when I raise the "transmit power" by 10×, the detection range barely grows?
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That's the most interesting thing about radar. The received echo power is inversely proportional to the fourth power of range. On the way out from the radar to the target, the wave spreads spherically and thins as 1/R². When the wave scattered by the target comes back, it thins as 1/R² again. Out and back, that's 1/R⁴. So detection range grows only as the fourth root of transmit power — to double the range you need sixteen times the power. That cliff-like curve in the "received power vs range" chart below is exactly this.
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Sixteen times! So is that fourth root also why "stealth aircraft" are hard for radar to see?
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Exactly that. RCS — radar cross section — expresses how easily a target reflects radio waves, as an area, and detection range scales with the fourth root of RCS. If an ordinary fighter has an RCS of a few m², shaping the airframe to deflect the wave elsewhere and adding radar-absorbent material can cut the RCS to 1/16. That only halves the opposing radar's detection range. Put the other way round, even a small drop in RCS reliably shrinks the detection range. That's why stealth design is so obsessed with shaping the airframe to chip away at RCS.
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When I change the frequency, the display switches between "X-band", "S-band" and so on. What's the difference?
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Those are radar bands — names given to ranges of transmit frequency. The higher the frequency, the shorter the wavelength. A short wavelength lets you use a smaller antenna and gives better angular resolution — that's why airborne fire-control and weather radars often use X-band (8-12 GHz). On the other hand, the longer wavelengths of L- and S-band suffer less attenuation in rain and the atmosphere and reach further, so they suit long-range search like an airport's air-traffic-control radar. You pick the band by balancing wavelength, detection range and antenna size for the job.
Frequently Asked Questions
The radar range equation gives the maximum range R_max at which a radar can detect a target, from transmit power, antenna gain, frequency (wavelength), the target's RCS (radar cross section), the receiver's minimum detectable signal and system loss. For a monostatic radar — one that uses the same antenna to transmit and receive — it is R_max = [P_t·G²·λ²·σ / ((4π)³·S_min·L)]^(1/4). Because the wave spreads twice, once outbound and once on the return, the received echo power falls off as the inverse fourth power of range.
The received echo power is inversely proportional to the fourth power of range (R⁴). The wave spreads spherically on its way from the radar to the target, thinning as 1/R², and the scattered wave thins again as 1/R² on its way back. Therefore R_max is proportional to the fourth root of the received power: R_max ∝ (P_t·G²·σ)^(1/4). Doubling the detection range requires sixteen times the transmit power — which is why simply turning up the output to see further is so expensive.
RCS σ is an effective area (m²) that describes how strongly a target reflects energy back to the radar. The received echo power is proportional to σ, but the detection range R_max scales with the fourth root of σ. So even cutting the RCS to 1/16 with shaping and radar-absorbent material only halves the opposing radar's detection range. Conversely, any reduction in RCS reliably shortens the detection range, which is why RCS reduction is the central tool of stealth design.
The radar band is the name of the transmit-frequency range: below 2 GHz is L-band, 2-4 GHz is S-band, 4-8 GHz is C-band, 8-12 GHz is X-band, 12-18 GHz is Ku-band, 18-27 GHz is K-band and above 27 GHz is Ka-band. Higher frequencies have shorter wavelengths, allowing smaller antennas and better angular resolution, but suffer more attenuation from rain and the atmosphere. Long-range search such as air-traffic control uses L- and S-band, while airborne fire-control and weather radars favour X-band.
Real-World Applications
Air-traffic-control radar: A primary surveillance radar (PSR) searching for aircraft over an airport and its surrounding airspace is a direct design target of the radar range equation. To cover 100-400 km it uses megawatt-class peak power and a large antenna, and chooses the rain-tolerant longer wavelengths of L- and S-band. Confirming in this tool that R_max grows only as the fourth root of transmit power gives an intuitive sense of why control radars become such enormous installations.
Weather radar: Treating raindrops and snow as "targets", a weather radar derives the echo intensity from the total RCS of the precipitation particles and converts it to a rainfall rate. The difference from an aircraft target is that many small scatterers are distributed within a volume, but the basic structure — received power falling as 1/R⁴ — is the same. Catching faint, distant precipitation hinges on how low the receiver's minimum detectable signal S_min can be made.
Military and air-defence systems, and stealth design: The relationship between an air-defence radar and an aircraft is precisely a contest over the radar range equation. The radar side tries to extend detection range by raising transmit power, antenna gain and receiver sensitivity; the aircraft side tries to shrink it by cutting RCS with shaping and radar-absorbent material. Because detection range works through the fourth root of RCS, even a small RCS reduction brings a large benefit to the defending side.
Automotive millimetre-wave radar and marine radar: An automotive collision-avoidance radar uses the 76-81 GHz band (above this tool's upper limit) to view short ranges of tens to hundreds of metres at high resolution. Marine radars use X- or S-band to pick up other ships, coastlines and buoys. In every case the design process works backwards from the range to be detected, the target RCS and the acceptable antenna size, using the radar range equation to set transmit power and frequency.
Common Misconceptions and Pitfalls
The biggest misconception is the belief that "detection range grows in proportion to transmit power". Because the received echo power is inversely proportional to the fourth power of range, R_max scales only with the fourth root of transmit power P_t. Doubling the power extends the detection range by about 1.19×; to double the range you need sixteen times the power. Move the transmit-power slider in this tool and look at the flat curve in the "max detection range vs transmit power" chart, and you'll see how poor a deal raising the output is. In practice, gaining antenna gain, receiver sensitivity or signal-processing integration gain is far more efficient than raising transmit power.
Next is the oversimplification that "RCS is a single number intrinsic to the airframe". RCS varies by orders of magnitude with the viewing direction (aspect angle), the frequency and the polarisation. The same aircraft has a very different RCS seen head-on versus seen broadside, and the σ this tool handles is just one representative point. Real stealth design performs direction-dependent optimisation — thoroughly cutting the head-on RCS while compromising on other directions. When you talk about detection range with a single σ, you need to keep that assumption in mind.
Finally, the overconfidence that "this radar range equation alone fully predicts detection performance". The formula in this tool is the most basic form and does not include atmospheric attenuation, ground or sea clutter, integration of receiver noise, target fluctuation (the Swerling models), or the probabilities of detection and false alarm. The actual detection range depends far more on environmental conditions and the quality of the signal processing than on the R_max computed here. This tool is an educational model for intuitively grasping "how each parameter affects detection range"; guaranteeing the performance of a real radar requires the full radar range equation referenced to signal-to-noise ratio (SNR) together with statistical detection theory.
How to Use
Select transmitter power (ptNum: 1–100 kW) via slider; higher power extends detection range proportionally.
Set antenna gain (gNum: 10–50 dB) and operating frequency (freqNum: 1–40 GHz) to define the radar band (L, S, C, X, Ku, K).
Input target radar cross-section (rcsNum: 0.1–1000 m²) representing reflectivity; jet fighters ~5 m², ships ~100 m², weather systems ~10 m².
Observe R_max (km), wavelength λ, echo power at maximum range (dBm), and detection class (fighter/maritime/weather).
Worked Example
Air traffic control radar: Pt=50 kW, G=35 dB (antenna gain), f=5 GHz (C-band, λ=6 cm), target RCS=8 m² (medium transport aircraft). Solver computes R_max≈185 km, echo power at R_max≈−88 dBm (above typical receiver sensitivity of −95 dBm), detection class: commercial aircraft. Reducing RCS to 1 m² (stealth signature) drops range to ~82 km; doubling power to 100 kW extends range to ~233 km.
Practical Notes
Frequency choice dictates weather effects: X-band (10 GHz) suffers rain attenuation >0.5 dB/km; L-band (1.5 GHz) penetrates weather, used for long-range air defense.
RCS values scale with frequency; a 10 m² target at 3 GHz may exhibit 20 m² signature at 10 GHz due to resonance effects.
Echo power below receiver noise floor (−100 to −120 dBm typical) yields no detection; integrate multiple pulses (pulse integration) to lower effective threshold by 10–20 dB.
Monostatic radar assumes transmitter and receiver colocated; bistatic systems (separated antennas) increase effective RCS by 4–8× due to geometry.