The inverse square calculator applies the inverse square law, which says that the intensity of anything spreading out from a point source falls with the square of the distance. Enter any three of I₁, d₁, I₂, and d₂ and the calculator solves for the fourth, then reports the intensity ratio, the percentage change, the change in decibels, and the change in photographic stops. Presets cover light in lux, sound intensity, gravitational force, and radiation dose rate. A second mode works out the plain mathematical inverse of a square, 1/x², for any number.
Inverse Square Calculator
Returns 1/x², x², and √x so you can compare the three "inverse square" meanings.
Result
Calculated I₂ = I₁ × (d₁ / d₂)²
100 lux
Intensity ratio
1/4 (0.25)
Distance ratio
2×
Change
−75%
Level change
−6.0206 dB
Copy answer
100
How the Energy Spreads Out
The same rays that light one square at d₁ cover (d₂/d₁)² squares at d₂, so each square gets less. Brightness shows intensity.
Intensity vs Distance Graph
The 1/d² curve through your two points. Hover to read the intensity at any distance.
What is the Inverse Square Law?
The inverse square law states that a quantity radiating from a point source has an intensity inversely proportional to the square of the distance from that source: I ∝ 1/d².
The reason is geometry. Energy leaving a point spreads over the surface of an expanding sphere. A sphere's surface area is 4πr², so at twice the radius the same energy covers four times the area, and every square metre receives a quarter as much.
Quantities that follow the law include:
- Light from a bulb, a star, or a flash (illuminance in lux).
- Sound from a small source in open air (intensity in W/m²).
- Gravity and electric force between point masses and charges (Newton's and Coulomb's laws).
- Ionising radiation from a small radioactive source (dose rate in µSv/h).
Interactive: Same Energy, Bigger Sphere
The source emits the same 60 rays at every radius. Drag r and count how thinly they are spread over the growing circle.
Surface area
4π (12.57)
Area ratio
1×
Intensity
100%
How to Calculate with the Inverse Square Law?
Use I₁ × d₁² = I₂ × d₂² and rearrange for the unknown. The product of intensity and distance squared stays constant for a given source.
- Write down the known pair. Measure or look up the intensity I₁ at a reference distance d₁.
- Form the distance ratio. Divide the old distance by the new one: d₁/d₂. Both distances must use the same unit.
- Square it. Squaring the ratio gives the factor the intensity changes by.
- Multiply. I₂ = I₁ × (d₁/d₂)². For an unknown distance, use d₂ = d₁ × √(I₁/I₂) instead.
Example: a lamp gives 400 lux at 1 m. At 2.5 m it gives 400 × (1/2.5)² = 400 × 0.16 = 64 lux.
Interactive: Step-by-Step Solution
Change the reference intensity and the two distances to watch each step update.
How to Use This Inverse Square Calculator?
Choose what you are measuring, pick the unknown in "Solve for," fill in the other three values, and press calculate.
1
Find a new intensity
Solve for I₂ when you know the reading at one distance and want it at another, such as lux on a desk after raising a lamp.
2
Find a safe or target distance
Solve for d₂ to find how far away you must be for a target value, such as a radiation dose limit or a sound level.
3
Reciprocal of a square
Switch to the 1/x² tab for the pure maths version: the reciprocal of x squared, alongside x² and √x.
The result panel also gives the change in decibels, useful for sound, and in stops, useful for photography. Distances can be in any unit as long as d₁ and d₂ use the same one.
Inverse Square Law Table
How intensity changes as you multiply the distance. Click a row to load it into the calculator with I₁ = 100 at d₁ = 1.
| Distance × k | Intensity fraction 1/k² | Remaining | Change in dB | Change in stops |
|---|---|---|---|---|
| 0.5× | 4 | 400% | +6.02 dB | +2 |
| 1× | 1 | 100% | 0 dB | 0 |
| √2 ≈ 1.41× | 1/2 | 50% | −3.01 dB | −1 |
| 2× | 1/4 | 25% | −6.02 dB | −2 |
| 3× | 1/9 | 11.1% | −9.54 dB | −3.17 |
| 4× | 1/16 | 6.25% | −12.04 dB | −4 |
| 5× | 1/25 | 4% | −13.98 dB | −4.64 |
| 10× | 1/100 | 1% | −20 dB | −6.64 |
| 100× | 1/10,000 | 0.01% | −40 dB | −13.29 |
Inverse Square Law for Light, Sound, Gravity and Radiation
The same 1/d² rule shows up across physics, but each field states it in its own units.
- Light: illuminance E = I/d², where I is luminous intensity in candela. Photographers count the change in stops: each halving of light is one stop.
- Sound: the sound pressure level drops by 20·log₁₀(d₂/d₁) dB, which is 6 dB per doubling of distance in free field.
- Gravity: g(h) = g₀ × (R/(R + h))², with R = 6371 km for Earth. The ISS at 420 km still feels about 88% of surface gravity.
- Radiation: dose rate falls with 1/d², so distance is one of the three pillars of radiation safety, along with time and shielding.
Interactive: Real-World Examples
Pick a field and drag the distance. The value at the reference distance stays fixed.
Inverse Square vs Inverse and Linear Relationships
An inverse square relationship falls much faster than a simple inverse (1/d) relationship. At 10 times the distance, 1/d leaves 10% while 1/d² leaves only 1%.
Telling them apart matters:
- 1/d: energy spreading from a long line, such as a busy road or a fluorescent tube, spreads over a cylinder, not a sphere.
- 1/d²: energy from a point spreads over a sphere. This is the inverse square law.
- 1/d³: fields of dipoles, such as a bar magnet far away, fall off faster still.
On a log-log plot each law is a straight line whose slope is the exponent: −1, −2, or −3. Plotting measured data that way is the quickest test of which law it follows.
Interactive: Compare Fall-Off Curves
Toggle the curves and switch to a log-log scale to see each law become a straight line.
1/d 1/d² 1/d³
The Inverse of a Square Number: 1/x²
In pure maths, the inverse of a square usually means its reciprocal: 1/x², also written x−2. Picture a unit square cut into an x-by-x grid. There are x² small squares, and each one is 1/x² of the whole.
- 1/2² = 1/4 = 0.25
- 1/3² = 1/9 ≈ 0.1111
- 1/10² = 1/100 = 0.01
- 1/0.5² = 1/0.25 = 4, since squaring a number below 1 makes it smaller and its reciprocal larger.
Don't confuse this with the inverse function of squaring, which is the square root. Squaring 3 gives 9, the square root of 9 gives back 3, and the reciprocal of 9 is 1/9.
Interactive: One Square out of x²
Drag x to slice the unit square into an x × x grid. The highlighted cell is 1/x² of the whole.
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Answers about distance, intensity, decibels, and 1/x²
Doubling the distance cuts the intensity to one quarter. The same energy spreads over a sphere with four times the surface area, because area grows with the square of the radius. Tripling the distance leaves one ninth, and ten times the distance leaves one hundredth.
I₁ × d₁² = I₂ × d₂², or equivalently I₂ = I₁ × (d₁ / d₂)². I is the intensity (or force, or dose rate) and d is the distance from the source. Rearranging gives d₂ = d₁ × √(I₁ / I₂) when you need the distance for a target intensity.
About 6 dB quieter in open air. Intensity drops to one quarter, and 10 × log₁₀(1/4) = −6.02 dB. Indoors, reflections from walls add energy back, so the real drop is often smaller than the inverse square law predicts.
Yes. Newton's law of gravitation, F = G·m₁·m₂ / r², is an inverse square law. At twice the distance from Earth's centre (one Earth radius above the surface), gravity is one quarter as strong, about 2.45 m/s² instead of 9.81 m/s².
It applies only to a point source radiating evenly in all directions, with nothing absorbing or focusing the energy. Close to large sources (such as a softbox or a long fluorescent tube), with lasers or spotlights, in fog or water that absorbs energy, and in rooms with strong echoes, the drop-off is different. A common rule is to measure at least five times the source size away.
No. The reciprocal of a square is 1/x², while the inverse operation of squaring is the square root √x. For x = 4, the square is 16, the reciprocal of the square is 1/16 = 0.0625, and the square root is 2. This calculator's second tab shows all three.
Moving a light to 1.4 times the distance halves its brightness, which is one f-stop. Doubling the distance loses two stops. Photographers use this to set flash power and to control how quickly light falls off across a subject or background.