What this calculator does
The inverse square law describes how the intensity of something spreading out from a point source falls off with distance: intensity is proportional to one over distance squared. Double the distance from a light, a speaker or a radiation source, and the intensity does not halve, it drops to a quarter, because the same energy is spread over four times the area.
The law shows up across physics because it follows purely from geometry: energy radiating outward from a point spreads over the surface of an ever-larger sphere, and the surface area of a sphere grows with the square of its radius. Light intensity, sound intensity, radiation exposure and gravitational force all obey the same 1/d² relationship for exactly this reason, even though the underlying physics of each is different.
The formula
Given the intensity at one distance, the intensity at a second distance equals the first intensity multiplied by the square of the ratio of the two distances (first distance over second distance). Moving further away divides intensity by the square of how much further; moving closer multiplies it by the square of how much closer.
| Term | Meaning |
|---|---|
| I₁, I₂ | Intensity at the first and second distances. |
| d₁, d₂ | The first and second distances from the source. |
The inputs explained
| Field | What to enter |
|---|---|
| Intensity at first distance | The known intensity at the first distance, in whatever units it is measured (lux, watts per square metre, decibels are handled differently, and so on). |
| First distance (m) | The distance at which that first intensity was measured. |
| Second distance (m) | The second distance to find the intensity at. |
When to use it
Photography and lighting
Moving a light source further from a subject predictably reduces exposure according to the inverse square law, which is why photographers halve or double distance deliberately to control light falloff.
Radiation safety
Increasing distance from a radiation source is one of the most effective ways to reduce exposure, and the inverse square law quantifies exactly how much a given increase in distance helps.
Acoustic and antenna design
Sound intensity and radio signal strength both fall off with the square of distance from the source in open space, which is a starting point for estimating coverage or audibility at range.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does intensity fall off as distance increases?
The same source intensity at 2 metres, evaluated at a range of second distances.
Questions
Does the inverse square law apply to any kind of intensity?
It applies to anything spreading uniformly in all directions from a point source through open space: light, sound, radiation, gravitational force. It does not hold for a focused beam (like a laser or a directional speaker) or when something absorbs or blocks part of the path.
Why does doubling distance quarter the intensity rather than halve it?
Because the energy spreads over the surface of a sphere, and a sphere’s surface area grows with the square of its radius. Doubling the radius quadruples the surface area the same energy is spread across, so the intensity at any point on that surface drops to a quarter.
Is this the same formula as Newton’s law of gravitation?
They share the same 1/d² shape, but Newton’s law also includes the two masses and the gravitational constant, giving an actual force in newtons rather than a general intensity ratio. This calculator handles the general relationship; a dedicated gravitational force calculator handles the full formula.
What if the source is not really a point?
The law is exact only for a true point source or far enough from an extended source that it behaves like one. Close to a large or spread-out source, the relationship is more complicated.
For the full gravitational force formula with both masses and the gravitational constant, see the Newton's law of gravitation calculator.