Inverse Square Law Calculator - Intensity vs Distance From a Point Source

An inverse square law calculator turns one measured intensity-and-distance pair into the intensity at any other distance from the same point source, using the geometric dilution formula I1/I2 = r2²/r1².

• • Light falloff at a new distance: Take a lux reading near a lamp and predict the illuminance a few meters away without remeasuring.
• • Solar irradiance at planetary orbits: Convert the 1361 W/m² value at 1 AU into the irradiance at Mars, Venus, or any other planet's orbit.
• • Radiation safety distance: Estimate how far you have to move from a sealed radioactive source to drop its dose rate by a chosen factor.
• • Sound level at a new distance: Adjust a measured SPL reading for the 1/r² geometric dilution that applies outdoors to a point-like sound source.

The inverse square law applies whenever a quantity spreads radially from a point source in three-dimensional space, because the area of a sphere of radius r is 4πr². Doubling the radius spreads the same total output over four times the area, so the per-area intensity drops to one quarter. Tripling the radius spreads the output over nine times the area, so the intensity drops to one ninth.

Anything that propagates outward from a point, like light from a small bulb, the gravitational pull of a planet, the Coulomb field of a point charge, or the acoustic power of a small speaker, follows this same geometric rule. The same equation works for all of them because the underlying geometry is the same; only the units of intensity change.

When the same point source is also radiating thermally, the Blackbody Radiation Calculator converts the surface temperature into the Stefan-Boltzmann radiant exitance that the inverse square law then dilutes at distance.

The calculator takes a reference intensity and distance, reads the target distance (or target intensity) that you supply, and applies the geometric dilution formula I1/I2 = r2²/r1² to solve for the missing value.

• I1: Reference intensity at the reference distance. Use the same unit system you will use for I2 (for example W/m² for irradiance, lux for illuminance, or any consistent arbitrary unit).
• r1: Reference distance from the point source where I1 was measured. Use the same length unit you will use for r2 (meters, astronomical units, or any consistent length).
• I2: Intensity at the target distance r2. Leave this field blank to solve for it, or fill it in to back-solve for r2 instead.
• r2: Target distance from the point source. Leave this field blank if you are solving for I2; otherwise set it to the distance at which you want to know I2.

Pick a solve mode by leaving one of the four fields empty. With three values known, the missing value falls out of the same formula no matter which three they are, and the dilution factor (r1/r2)² makes it easy to see whether the missing intensity is smaller or larger than the reference.

If you set I1 = 1361 W/m² at r1 = 1 AU and r2 = 1.524 AU, the dilution factor is (1/1.524)² ≈ 0.4305 and I2 = 1361 × 0.4305 ≈ 585.9 W/m².

According to HyperPhysics, intensity scales as 1/r^2 because it spreads over the area of a sphere of radius r

If the inverse square law is being used to estimate how much signal reaches a telescope aperture, the Angular Resolution Calculator uses that same wavelength to set the smallest angular separation two point sources can still be resolved.

Four ideas show up every time you work with an inverse square law problem: the geometry of a sphere, the meaning of point source, the distinction between total output and per-area intensity, and the geometric dilution constant.

The 1/r² rule is geometric, not physical. Once a quantity spreads radially through three-dimensional space, the area it must cover grows quadratically with distance, so the per-area value must shrink by the same factor.

Kepler's third law comes from the same 1/r² gravitational force that drives this calculator, and the Orbital Period Calculator turns a planet's semi-major axis into its orbital period so the geometric dilution plays out over time.

Pick what you want to solve for, type the three values you already know, and read the missing value plus the dilution factor in the result panel.

1. 1 Choose the solve mode: Use the first dropdown to pick whether you want to solve for intensity (I2) or distance (r2).
2. 2 Enter the reference intensity I1: Type the measured or tabulated intensity at the reference distance (for example 1361 for solar irradiance at 1 AU in W/m²).
3. 3 Enter the reference distance r1: Type the distance from the point source to where I1 was measured. Use the same length unit you will use for r2.
4. 4 Enter the target value: If you picked 'Solve for intensity', enter r2. If you picked 'Solve for distance', enter I2.
5. 5 Read the result panel: The panel shows the missing intensity or distance, the dilution factor (r1/r2)², the intensity ratio, and the geometric dilution constant I·r² for both reference and target points.
6. 6 Verify with the dilution constant: Compare I1·r1² with I2·r2². They should match within rounding if the inputs describe the same source and the same unit system.

A laboratory measures 80 W/m² at 1 m from a small outdoor speaker. To find the stand-off where the linear acoustic intensity has dropped to 5 W/m², switch to solve for distance, enter I1 = 80, r1 = 1, and I2 = 5. The calculator returns r2 = 4 m, the textbook 1/r² doubling rule.

For a sound problem where the inverse square law gives the geometric dilution, the Harmonic Wave Equation Calculator converts the source frequency into wavelength so the same distance can be cross-checked against any near-field correction.

A dedicated inverse square law calculator handles the unit bookkeeping and the direction-of-solve logic that slows down 1/r² problems by hand.

• • Two solve directions in one panel: Switch between solving for intensity at a new distance and solving for the distance that gives a target intensity without retyping the formula.
• • Mixed-unit safety: All distances are treated as ratios, so r1 and r2 in different length units still produce a correct dilution factor when both inputs use the same unit system.
• • Domain-agnostic: Use the same calculator for solar irradiance, light bulb illuminance, acoustic SPL scaling, or gravitational field strength, because the underlying 1/r² relationship is identical.
• • Sanity-checkable result: The reported I·r² constant for the reference and target points lets you verify that the two measurements describe the same source to within rounding.

Use the same length unit for r1 and r2 in every problem. If r1 is in meters and r2 is in feet, the calculator treats them as the same unit and silently gives a wrong answer; convert first.

When the radial quantity is gravitational force rather than light intensity, the Newton's Laws Calculator evaluates F = G m1 m2 / r² so the inverse square law result can be paired with the actual force between two masses.

Four things change what an inverse-square-law result really means. Review them before you trust the number on a report.

• • The calculator assumes an isotropic point source and a non-absorbing medium. Real sources and real media deviate, so the reported I·r² constant should be checked between two measured points before extrapolation.
• • Intensity units are intentionally left unitless in the form. The user must keep I1 and I2 in the same unit system.

If the medium temperature, density, or composition changes between the reference and target distance, expect the measured constant to drift. Treat the calculator as a geometric tool and apply atmospheric corrections separately.

According to Britannica, inverse-square behavior applies to gravity, light, electrostatics, and any radially spreading quantity

According to NASA Science - Solar System Exploration, solar irradiance at 1 AU is 1361 W/m^2 and the Earth-Sun distance is 1 AU

Once a real medium absorbs or scatters the radiated quantity, the falloff is faster than 1/r², and the Attenuation Calculator applies the Beer-Lambert exponential decay law so you can combine that absorption loss with the geometric dilution for a complete picture of intensity drop in air, water, or shielding.