Inverse Square Law Calculator

The inverse square law says the intensity of a point source falls with the square of the distance: I is proportional to 1/d^2, or I1/I2 = (d2/d1)^2. Physically, the energy spreads over a sphere whose area grows as 4*pi*r^2, so doubling the radius quadruples the area and each patch receives one quarter of the intensity.

It drops to exactly one quarter. For example, 100 lux at 2 m becomes 25 lux at 4 m, and 500 lux at 2 m becomes 125 lux at 4 m, because (2/4)^2 = 1/4. Triple the distance and you get one ninth. The table below shows the full doubling ladder: 100, 25, 6.25, 1.5625, 0.390625.

Use I1/I2 = (d2/d1)^2. If you know the intensity I1 at distance d1, the intensity at d2 is I2 = I1*(d1/d2)^2. The solver above works all four ways: give it any three of I1, I2, d1, d2 and it finds the fourth, and it always shows the substituted formula so you can check the arithmetic.

6.02 dB of sound pressure level. SPL follows 20*log10(d1/d2), and 20*log10(1/2) = -6.02 dB. So 90 dB at 1 m becomes 83.98 dB at 2 m, then 77.96 dB at 4 m. Each doubling removes another 6.02 dB.

Because they measure different things. Sound pressure level (SPL, in dB) is defined from pressure, an amplitude quantity, so the log gets a factor of 20. Acoustic intensity is a power quantity, so its level uses 10*log10(I1/I2). A 6.02 dB SPL drop equals a 6.02 dB intensity-level drop too, but the formulas differ: 20 for pressure, 10 for power. This calculator labels which one it uses.

Yes. Newton's law g = G*M/r^2 is an inverse square law, so g1/g2 = (r2/r1)^2. At twice the distance from Earth's center, gravity is one quarter: 9.81 m/s^2 at the surface (6,371 km from the center) becomes 2.4525 m/s^2 at 12,742 km. Remember r is measured from the center of mass, not from the ground.

Dose rate from a point source also follows 1/d^2: double the distance, quarter the dose. For example, 10 uSv/h at 1 m becomes 1.11 uSv/h at 3 m, since (1/3)^2 = 1/9. This is the distance leg of ALARA (time, distance, shielding). Informational only, not medical or safety advice.

It assumes a point source radiating uniformly into free space. It breaks down in the near field (closer than a few times the source size), for extended or line sources (a long fluorescent tube falls closer to 1/d), when air or materials absorb or scatter the energy, and for collimated beams like lasers that barely spread.

This is a general multi-domain solver for I proportional to 1/d^2 (light, sound, gravity, radiation). Our Lighting Unit Converter is the photometry specialist: lux, lumens, foot-candles and candela conversion, watts-to-lumens by bulb type, and a room lighting planner. Our Grow Light Lab covers hanging distance and PPFD/DLI for grow lights. Use the right tool for the job.

Yes, the solver works four ways. Pick what to solve for: for example, with 100 at 2 m and a target of 25, the required distance is d2 = 2*sqrt(100/25) = 4 m. In the sound preset you can likewise solve for the distance that gives a target dB level.