Inverse Square Law Calculator (intensity vs distance for light, sound, radiation)

Understanding the inverse square law calculator — concept and principle

This calculator applies the inverse square law to find how the intensity of light, sound, radiation or any other energy spreading out from a small source changes with distance. It also gives the ratio, the change in decibels, the distances for half and one tenth of the intensity, and the source power. It is used in photography, radiation safety, acoustics and physics classes.

The same light covers a patch 3 × 3 = 9 times larger, so each square gets one ninth: 100 becomes 11.11.

How it is calculated

I₂ = I₁ × (r₁ ÷ r₂)²

Change in level = 10 log₁₀(I₂ ÷ I₁) dB

Distance for half the intensity = r₁ × √2; for one tenth = r₁ × √10

Source power (W, if I is in W/m² and r in m) = 4π r₁² I₁

The two distances must be in the same unit; the intensity can be in any unit.

The principle

A small source that radiates equally in all directions spreads its power over the surface of a growing sphere. A sphere's area is 4πr², so at twice the distance the same power covers four times the area and the intensity falls to a quarter. Because the power cancels when two distances are compared, you do not need to know it. The law holds when the distance is large compared with the source and nothing on the way absorbs, reflects or focuses the energy.

Worked example

Default inputs: intensity 100 at 1 (any unit), second distance 3.

Ratio = (1 ÷ 3)² = 0.1111.
I₂ = 100 × 0.1111 = 11.11.
Change = 10 log₁₀(0.1111) = −9.54 dB.
Half intensity at 1 × √2 = 1.414; one tenth at 1 × √10 = 3.162.
If the units were W/m² and m, the source power would be 4π × 1² × 100 = 1,257 W.

Each point is the calculator re-run with only “Second distance (same unit)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Second distance (same unit)	Intensity at the second distance
0.75	177.8
1.5	44.44
2.25	19.75
3	11.11
3.75	7.111
4.5	4.938
6	2.778

How to read the result

Every doubling of distance cuts intensity to a quarter (−6 dB), and every tenfold increase to one hundredth (−20 dB). That is why stepping back from a radiation source is so effective and why a flash must be much stronger to light a subject twice as far away. Near large sources, long tubes or in rooms with reflections, real intensity falls more slowly.

Common mistakes

Using the ratio r₁/r₂ without squaring it.
Applying the law to lasers, spotlights or line sources such as long tubes and busy roads.
Measuring distance from the wrong point; use the effective centre of the source.
Using it close to a source that is large compared with the distance.

Keep in mind: Radiation and sound exposure limits depend on more than distance. Follow the safety rules that apply to your source.

Frequently asked questions

Why does doubling distance give a quarter, not a half?

The energy spreads over an area that grows with the square of the distance.

Does sound follow the inverse square law?

In open air away from reflecting surfaces, approximately; indoors, reflections keep levels higher.

How many decibels is a halving of intensity?

About −3 dB.

Does gravity follow the same law?

Yes. The gravitational force from a mass also falls with the square of the distance.

Sources: OpenStax, University Physics Volume 1 (2016), section 17.3 "Sound Intensity"

Checked 9 Nov 2026. OpenStax, University Physics Volume 1 (2016), §17.3 “Sound Intensity” (I = P/A with A = 4πr² for a spherical wave; I₂ = I₁ (r₁/r₂)², intensity falls as the inverse square of distance).

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