Inverse Square Law Calculator

What Is the Inverse Square Law Calculator?

How to use the inverse square law calculator

1. Choose the unknown. The Solve for menu opens on Intensity wanted, and the other three choices are Distance wanted, Known intensity and Distance of that reading. The source power is never the unknown.
2. Type the known intensity. The reading you already have, in W/m^2, mW/m^2, kW/m^2 or uW/m^2, opening on 48 W/m^2 with presets at 48, 100 and 1000. The box accepts 1e-12 to 1e12 W/m^2.
3. Type the distance of that reading. How far it was taken from the source, measured from the centre — not the surface, not the housing. It takes m, cm, mm or km and opens on 0.5 m.
4. Type the distance wanted. Where you want the reading, from the same centre, opening on 3 m with presets at 1, 3 and 10 m. Tripling it divides the reading by nine.
5. Leave the power box alone unless you have it. It opens on 0, which means you do not know the source power, and the ratio form never needs it. Fill it in and the first reading beside the answer shows your own figure instead.
6. Read the answer. The headline carries the answer to six significant figures, trailing zeros dropped, in the unit that box was showing, and it switches to scientific notation below 0.0001 and at or above 10 million. Set the unit first, point Solve for at the target second and type your figures last — switching the target wipes every box back to its default and only the unit menus survive it.
7. Read the source power. P = 4πr²I at your second distance, and it is that power only if the source radiates equally in all directions, which is why the label says so.
8. Read the halving and doubling distances. Your distance multiplied and divided by the square root of two. At 1 m those are 1.414 m and 70.711 cm.
9. Read what a slip in the distance costs. A 1 per cent error in r is a 2.0100 per cent error in I, and the excess over double is exactly the square of the error.
10. Open Show working. The steps give the relation, the boxes in SI base units, the substitution, the centre rule, the ratio and its square, the power behind the reading, the halving and doubling distances with the source’s own quick rules beside them, the price of a slip, the assumptions and every unit conversion.

This page exists for the one step that turns a reading at one distance into a reading at another. The inverse square law guide is the prose half of it, where the exponent is derived and the problems are worked. The P in the optional box is a total radiated power in watts, which is the power guide’s quantity rather than this page’s.

Two mistakes account for most wrong answers here. The first is measuring the distance from the surface of a source that has a size, instead of from its centre. The second is reaching for the source’s power: there is an optional box for it, but the relation cancels it and you will not need it.

Worked example: change one thing at a time

The table starts at the defaults, runs the unknown through all four modes on one state, asks the distance question, loads the anchor case with and without a power, walks every unit menu, and finally reaches the entries the calculator declines. Every answer and reading cell was taken from the running widget rather than worked out by hand. Where a cell and the tool ever part company, believe the tool.

Rows 1 to 4 are one state — 48 W/m^2 at 0.5 m, wanted at 3 m — with the Solve for menu in each of its four positions. All three reading columns are identical down all four rows, because the readings describe the state rather than the mode. That is the round trip in one block: the headline prints the far reading as 1.33333 W/m^2, and what rows 2 to 4 feed back is the box’s own ten-digit 1.3333333333, which returns 48 W/m^2 and 0.5 m as printed.

Rows 5 to 7 ask the three questions the page is for. Row 5 is the distance mode: 48 W/m^2 has fallen to 3 W/m^2 at 2 m, because sixteen times the reading is four times the distance. Rows 6 and 7 are the same anchor — a 100 W source read at 1 m and again at 2 m — with and without the optional power, and the headline 1.98944 W/m^2 is identical in both, which is the whole claim about the ratio form in one pair of rows.

Rows 8 to 10 are the ones worth keeping. At 1 m the power reading is 12.566 W, which is the sphere area at that distance written in watts, because 4πr² and the watts for one watt per square metre are the same number there; row 9 is the decade, ten times the distance for one hundredth of the reading, every time. Row 10 is the sanity check on an emitter whose power you do know — the same figure the Stefan-Boltzmann calculator would hand you for a hot surface — and a kilowatt radiated evenly reaches 1 W/m^2 at 8.92062 m.

Rows 11 and 12 are the honest rows, and they are this page’s own subject turned on itself. Typing the rounded 7.958 instead of the exact anchor gives 1.9895 W/m^2 rather than 1.98944, and feeding the rounded 1.333 back gives 47.988 W/m^2 rather than 48. The relation squares a ratio, so it squares a rounding too.

Rows 13 to 15 change nothing but the units. 48000 mW/m^2, 50 cm and 300 cm, and 0.1 kW all reproduce the figures they came from exactly, because every multiplier in the menus is an exact power of ten. Rows 16 to 22 are the refusals, and they are refusals on the input rather than tidied-up output: every intensity and every distance sits in a division or under a square root, so a zero, a negative or an out-of-range entry is declined in words rather than answered with an infinity or a NaN.

Formula and symbol reference

One relation does all the work: I1 · r1² = I2 · r2², which rearranges to I2 = I1 (r1 / r2)² going outward and to r2 = r1 (I1 / I2)^(1/2) going the other way. It comes from one fact about a sphere: the area is 4πr², so the intensity at r is I = P / (4πr²) and dividing two such readings cancels P. Everything else on this page is that relation rearranged, or a quantity derived from the state it describes.

The unit of the two intensity boxes is the watt per square metre, which the SI units guide derives from the base units. Reading the relation backwards gives P = 4πr²I, which is the first reading beside the answer and the only place the power appears at all. Writing the law as I = P / (4πr²) needs the power; writing it as a ratio does not, and that is the difference the page is built on.

Two symbols on this page are not boxes. r2 / r1 is the distance ratio whose square is the whole answer, and R / s is the source radius over the distance from its surface, which only matters when you measure from the wrong place. Neither appears in the widget, and both appear in the working.

The physics: why the exponent is two, and what sets it

Whatever is radiated outward from a small source has to cross every sphere drawn around it, and nothing is lost in between. The sphere at r has an area of 4πr², so the same quantity is spread over four times the area at twice the distance. The exponent is not a property of light, or of sound, or of a force — it is a property of the surface the quantity spreads over.

That is the one idea worth carrying away, and it is also the reason the exponent sometimes is not two. A long line source spreads over a cylinder whose area is 2πrL, which grows as r, so the intensity goes as 1/r; a flat sheet spreads over a fixed area and does not fall off at all. A radar echo crosses a sphere twice, out and back, and goes as 1/r^4.

Because the exponent comes from geometry rather than from the quantity, the same 1/r² turns up in places that otherwise share nothing. It is the shape of the gravitational force law and of the electrostatic force law, and each of those needs its own pair of quantities — two masses, or two charges — that this page does not ask for. Naming them is this page’s business; computing them is theirs.

One pedantic point is worth a clause. The exponent depends on which quantity you mean: the source notes that for sound the pressure goes as 1/r while the intensity, which is that quantity multiplied by the particle velocity, goes as 1/r². Same source, same distance, two exponents — and the sound-level arithmetic that follows from it belongs to the decibel formula guide.

Where this arithmetic breaks down

The arithmetic here is exact to the last digit a double can hold. What goes wrong is physical, and one item on this list matters more than all the others.

Measuring from the wrong place costs a factor of four

This is the one to read. The law is exact in r taken from the centre of a spherically symmetric source. Take s from the surface instead and square that, and the reading you predict is too high by exactly (1 + R/s)²: 4.0000× one radius out, 1.4400× at five radii, 1.2100× at ten, 1.0201× at a hundred. The one per cent line sits at 200.4988 source radii, so a bench measurement is almost always over it.

An extended source is not a point source

This is a different error from the one above, and the two corrections must never be applied to each other. The source’s own rule is that when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1 per cent, and it adds the sharper warning that the proportionality does not hold in practical situations unless source dimensions are much smaller than the distance. That rule is quoted here and not extended, because working it out properly needs a model of the source this page does not have.

A beam is not a point source

The source is explicit: for non-isotropic radiators such as parabolic antennas, headlights and lasers, the effective origin is located far behind the beam aperture. Point this page at a torch or a laser and the distance you measured is not the r the relation means. No figure is offered for any beam, because nothing here sources a beam divergence.

The medium absorbs and scatters, and the model has neither

The relation assumes there are no losses caused by absorption or scattering, which is the source’s own wording for an assumption rather than a finding. Real air, water, glass and shielding all take energy out on the way, so a real reading falls faster than 1/r². No coefficient is printed here for any of that; the gamma ray guide is where attenuation and shielding live.

Reflections put energy back across the sphere

The law counts the energy crossing one sphere once. In a room, a reflected path crosses it twice, so a reading close to a wall or in a small space is higher than this page predicts and stops falling off entirely once the reflected field dominates. That is said in words and not in numbers, because nothing here sources a reflection coefficient.

The wrong geometry gives the wrong exponent

A reader who assumes 1/r² for a fluorescent tube at close range is out by the whole difference between 1/r and 1/r² — at ten times the distance, a tenth against a hundredth. There is no geometry box on this page by design, so if the source is long, flat or an echo, this is the wrong page and the shape of the spreading surface is the thing to fix first.

Rounding a distance and then squaring it

The error in the reading is always worse than double the error in the distance, by exactly the square of it. One per cent becomes 2.0100, five per cent becomes 10.2500, ten per cent becomes 21.0000 and fifty per cent becomes 125.0000. That is why the two quick rules about 1.4 and 0.7 both overshoot by 2.0408163 per cent from a rounding of about one per cent in the distance.

The detector has to face the source and be small

The source defines the intensity as what passes through unit area directly facing the point source, so a tilted detector reads less and a detector comparable in size with r averages over a range of distances. Neither is corrected for here, and the quantity itself — what it means and how it is defined — belongs to the polarisation of light guide rather than to this tool.

A clean answer can be a coincidence of the numbers

The opening state falls by exactly 36 and the anchor by exactly 4 because their distance ratios are exactly 6 and exactly 2. Those are tidy because the inputs are tidy, not because the arithmetic is. Change 3 m to 3.1 m and every figure turns ragged while every identity still holds to the last digit.

The calculator measures nothing — you supply the numbers

Two distances and one reading go in, one relation is used, and the answer describes the situation you described. The case names on the buttons are shorthand for the figures behind them and are not claims about any particular lamp, speaker, antenna or emitter.

Where the inverse square law is actually used

Moving a meter and predicting the new reading

Load 48 W/m^2 at half a metre, read at three and the trade is on screen in one line: 1.33333 W/m^2, with 4.243 m and 2.121 m beside it for where it halves and doubles. One reading, one tape measure, and the second reading never needed the source at all.

Finding the distance that hits a target reading

Point Solve for at the distance wanted and the question becomes the one a survey actually asks. 48 W/m^2 at 0.5 m reaches 3 W/m^2 at 2 m, and the square root is what makes that feel gentle: every factor of one hundred in the reading is a factor of ten in the distance.

Backing out the power of a source you cannot open

One reading and one distance give the total radiated power, provided the thing radiates equally in all directions. 1 W/m^2 at 1 m is 12.566 W, and that number is the sphere area at one metre written in watts. The assumption is in the label because an inferred power without it is not a figure worth quoting.

Setting a lamp for a one-stop change

The halving and doubling distances are a photographer’s whole working set. The exact factors are the square root of two and its reciprocal, and the familiar 1.4 and 0.7 are roundings of them that leave 0.5102 and 2.0408 rather than a half and a double. Over one step that is invisible; over several it is not.

Checking a published fall-off claim

Two readings and two distances are enough to test whether something really falls off as 1/r²: multiply each reading by its distance squared and the two products should match. If they do not, the source is not a point, the medium is absorbing, or the distances were measured from the wrong place.

Telling an intensity from a level

If what you actually need is a sound level rather than an intensity, that is a logarithm away and it is a different tool: the decibel calculator takes an intensity and returns a level, and it has no distance field at all. Work the distance here first, then take the result across.

Recognising the same law somewhere else

Gravity and electrostatics are the two most famous instances of this shape, and each needs its own pair of quantities rather than two readings. The gravitational force calculator is the one to compare against: identical exponent, completely different inputs, and a constant that this page does not have because the ratio form cancels it.

Where to go next

For the argument this arithmetic supports — why the exponent is two, what the ratio form buys, what four different spreading surfaces do to it and why measuring from the wrong place costs a factor of four — read the inverse square law guide, which this tool is the calculating half of.

Four posts pick up the pieces this page uses but does not explain. The law of gravitation guide and the Coulomb law guide are the two instances with their own constants, the decibel formula guide is where a sound intensity becomes a level, and the physics formula sheet is the one-page reference.

Four tools sit next to this one. The gravitational force calculator and the Coulomb law calculator are the two instances, the decibel calculator turns an intensity into a level, and the Stefan-Boltzmann calculator gives the power a hot surface emits. The decibel simulator is the one live lab that moves a source and a listener.

The full physics lab library and the calculator index are open too.