Inverse Square Law Calculator — how intensity falls with distance from a point source

The inverse square law calculator above answers the single most common question in radiometry, photometry, acoustics and radiation protection: I measured this much at that distance, so what will I get somewhere else? The law is pure geometry. Anything radiated equally in all directions from a small source spreads over the surface of an expanding sphere, that surface grows as the square of the radius, and so the amount landing on each square metre falls as one over the square of the distance.

Arb Digital publishes free physics tools that make the assumptions visible rather than hiding them. The inverse square law is one of the most abused relationships in engineering, because it is exactly true for an idealised point source in empty space and quietly wrong for almost every real source measured close up. This page gives the arithmetic and then spends most of its length on where the arithmetic stops applying, because that is where the errors actually happen.

What This Inverse Square Law Calculator Does

The first mode scales a measurement. Enter what you read at one distance and where you want to know the value instead, and the tool applies I₂ = I₁ (d₁/d₂)². The second mode starts from the source itself, dividing the total output by the surface area of a sphere at the distance you are interested in. The quantity selector changes only the unit label; the geometry is the same for a lamp, a loudspeaker and a gamma source.

The supporting grid gives the four figures that make the result useful rather than merely numerical. The intensity ratio is the raw multiplier. The change in decibels is the same information on the scale that acousticians and radio engineers actually use, and it is worth internalising that a doubling of distance is always a six decibel drop regardless of what is radiating. The one-tenth distance is the range at which the reference intensity has fallen by an order of magnitude, which is often a more useful planning figure than any specific value. The sphere area at the new distance shows the geometry directly — it is the surface over which the same total output is now spread.

Degenerate cases are explained rather than allowed to produce infinities. A distance of zero would give an infinite intensity, which is the mathematical signature of the point-source idealisation breaking down rather than a physical prediction, and the tool says so. A zero intensity or zero source output reports zero with an explanation. Distances are treated as magnitudes, and a negative entry is rejected with a message rather than silently squared into a positive.

How to Use It

1. Measure both distances from the source's centre. For a compact source that is obvious. For a long fluorescent tube or a large panel it is not, and using the distance to the surface produces a badly wrong answer at close range.
2. Take your reference reading far enough out. A good rule of thumb is at least three times the largest dimension of the source. Inside that distance the source is not acting like a point and the law does not apply.
3. Use total output, not beam output, in the second mode. A reflector lamp concentrates its lumens into a cone, so the sphere calculation will understate the on-axis intensity substantially.
4. Keep both distances in metres. The ratio is what matters, so any consistent unit works, but mixing metres with feet gives an error of more than a factor of ten.
5. Check the decibel figure against your intuition. Six decibels per doubling is the signature of a point source. If your measurements show three decibels per doubling, you are dealing with a line source and this page is the wrong model.

The Formula: How the Inverse Square Law Is Calculated

A point source radiating a total power P equally in all directions spreads it over a sphere of area 4πd², so the intensity at distance d is I = P / (4πd²). Comparing two distances makes the source power cancel out, leaving I₁d₁² = I₂d₂², or I₂ = I₁(d₁/d₂)². HyperPhysics on the inverse square law shows the same geometry applied to gravity, electric fields, light, sound and radiation, which is a good reminder that the law is a statement about three-dimensional space rather than about any particular kind of energy.

Work the defaults through. A reading of 1,000 W/m² at 1 m, evaluated at 3 m, gives 1,000 × (1/3)² = 1,000 ÷ 9 = 111.1 W/m². The ratio is 0.1111, which in decibels is 10 log₁₀(0.1111) = −9.54 dB. The distance at which the reference intensity would fall to one tenth is 1 × √10 = 3.16 m, just beyond where we evaluated. The sphere at 3 m has an area of 4π × 9 = 113.1 m².

In source mode, a 100 W isotropic radiator at 3 m gives 100 ÷ 113.1 = 0.884 W/m². That figure is often startlingly small, and it is the correct answer for an omnidirectional source; the reason a 100 W lamp seems brighter than that is that it is not omnidirectional. NASA's classroom resource on the inverse square law of light covers the measurement exercise that demonstrates the relationship, and the astronomical distance-measurement technique that depends on it.

The Near Field Is Where This Law Goes Wrong

The single largest source of error with the inverse square law is applying it too close to the source. The derivation assumes the source is a point, and no real source is. At a distance comparable to the source's own size, different parts of the source are at meaningfully different distances from you, and the total falls off far more slowly than the square.

The limiting cases are worth knowing. An infinite line source — a long strip light, a busy motorway, a run of pipe — falls off as one over distance, not one over distance squared, which is three decibels per doubling rather than six. An infinite plane source does not fall off with distance at all in the near field, which is why standing further back from a large illuminated ceiling barely dims it. Real sources sit somewhere between these behaviours depending on where you stand, transitioning to true inverse square only once you are far enough away that the source subtends a small angle.

The practical rule most disciplines use is that the inverse square law is reliable beyond roughly three times the source's largest dimension, and increasingly reliable further out. Inside that, measure rather than calculate. This is also why a photographer's rule about moving a softbox closer produces a much sharper falloff than the arithmetic suggests: a large softbox close to a subject is behaving like a plane, not a point.

Absorption, Reflection and Why the Real World Falls Off Differently

The law describes geometric spreading only. It assumes the intervening medium neither absorbs nor scatters, and that nothing reflects energy back. Both assumptions fail routinely, and they fail in opposite directions.

Absorption makes the real falloff steeper than inverse square. Air absorbs high-frequency sound noticeably over long distances, water absorbs light strongly and selectively by wavelength, and any shielding material attenuates radiation exponentially with thickness on top of the geometric spreading. Over long paths the exponential absorption term dominates the geometric one entirely.

Reflection makes the falloff shallower. Indoors, sound reaches a listener both directly and via every surface in the room, and beyond a certain distance the reflected field dominates and the level stops dropping at all. That distance is the critical distance, and it is why measuring a loudspeaker indoors gives results that contradict the inverse square law past a couple of metres. The same applies to light in a white-walled room. Our reverberation time calculator deals with the reflected field that this page deliberately ignores, and the decibel calculator handles the logarithmic arithmetic of combining several sources or converting between ratios and levels.

Where This Sits Next to Our Other Falloff Tools

The boundary worth stating is this: this page handles the geometry of a point source in free space, and nothing else. Several tools on the site handle the same falloff dressed in the conventions of a particular field, and each adds something this one deliberately does not.

For radio links, the free space path loss calculator applies the same one-over-distance-squared spreading but expresses it in decibels against frequency, because an antenna's effective capture area depends on wavelength. For photometry, the luminous flux converter and the luminous intensity converter translate between lumens, candelas and lux, which is the unit bookkeeping that sits underneath this calculation rather than replacing it. For solar irradiance arriving at a panel, the solar panel calculator works from local irradiance figures rather than deriving them from the Sun's output. For thermal emission from a hot body, the blackbody radiation calculator gives the total output that this page would then spread over a sphere. Dose-rate units are converted by the radiation dose converter.

Common Mistakes to Avoid

• Measuring distance from the surface instead of the centre — for an extended source the two differ enough to change the answer by a large factor at close range.
• Applying the law inside the near field — closer than about three source diameters, a real source behaves like a line or a plane and falls off far more slowly than the square.
• Ignoring reflections indoors — past the critical distance in a room, the reflected field dominates and levels stop falling, which contradicts the calculation entirely.
• Using beam output as if it were total output — a reflector or a horn concentrates energy into a cone, so spreading its rated output over a full sphere understates the on-axis figure badly.
• Forgetting absorption over long paths — air, water and shielding attenuate exponentially on top of the geometric spreading, and over distance that term dominates.

Related Free Tools From Arb Digital

For the logarithmic side of the same problem, use the decibel calculator, and for radio propagation the free space path loss calculator. Photometric unit conversion lives on the luminous flux converter and the luminous intensity converter. For the reflected sound field that this page excludes, see the reverberation time calculator. Thermal emission is covered by the blackbody radiation calculator, incident solar energy by the solar panel calculator, and dose units by the radiation dose converter. Everything Arb Digital publishes is on the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and estimation use. It models geometric spreading from an idealised point source in free space with no absorption or reflection, publishes no exposure limits or safety thresholds of its own, and is not a basis for radiation protection, noise compliance or lighting certification decisions.