Inverse Square Calculator

The inverse square law says that intensity falls with the square of the distance from a point source: I ∝ 1/r². Double the distance and the intensity drops to 1/4; triple it and the intensity drops to 1/9.

The reason is geometry. A source sends its output out in all directions, so it spreads over the surface of a sphere of area 4πr². At twice the radius the sphere has 4 times the area, and each square metre gets 1/4 of the share.

• Light: a lamp gives 1/4 of the illuminance at 2 m that it gives at 1 m.
• Sound: each doubling of distance lowers the level by about 6 dB.
• Gravity and electric force: both follow F ∝ 1/r².
• Radiation: stepping from 1 m to 3 m cuts the dose rate to 1/9.

To find the new intensity, multiply the old intensity by the square of the distance ratio: I₂ = I₁ · (d₁/d₂)². For 800 lux at 2 m moved to 10 m:

1. Divide the distances: d₁/d₂ = 2/10 = 0.2.
2. Square the ratio: 0.2² = 0.04.
3. Multiply: I₂ = 800 × 0.04 = 32 lux.

Check: I₁ · d₁² = 800 × 4 = 3200 and I₂ · d₂² = 32 × 100 = 3200. The product I·d² stays the same at every distance.

The distances can be in any unit (m, ft, cm) as long as d₁ and d₂ use the same one. The intensity keeps the unit of I₁.

Pick what you want to find, type the known values, and the calculator applies the inverse square law as you type. The steps:

1. Choose New intensity to find I₂ at a new distance, Find distance to reach a target intensity, or Point source to get the intensity from a source output.
2. Type the values. Numbers like 3.828*10^26 and expressions like √2 work.
3. Read the answer, the distance and intensity ratios, the change in decibels and in light stops.
4. Open the spreading-rays diagram or the intensity graph, and the worked steps.

The result can be copied, shared as a link, printed or kept in the history list.

The inverse square law formula is I = P / (4πr²) for a point source, and I₂ = I₁ · (d₁/d₂)² between 2 distances.

• New intensity: I₂ = I₁ · (d₁/d₂)².
• Distance for a target: d₂ = d₁ · √(I₁/I₂).
• Constant product: I · d² = k, so I = k/d².
• In decibels: ΔL = 20 · log₁₀(d₁/d₂), about −6.02 dB per doubling.
• In light stops: each doubling of distance costs 2 stops (a factor of 4).

To find the distance for a target intensity, use d₂ = d₁ · √(I₁/I₂). For a source reading 100 units at 1 m, the distance where it reads 25:

1. Divide the intensities: I₁/I₂ = 100/25 = 4.
2. Take the square root: √4 = 2.
3. Multiply by d₁: d₂ = 1 × 2 = 2 m.

Cutting the intensity to 1/100 needs 10 times the distance, since √100 = 10. Raising it 4 times needs half the distance.

A point source with output P spreads it over a sphere of area 4πr², so the intensity at distance r is I = P / (4πr²). A 100 W source gives 100 / (4π × 2²) = 1.989 W/m² at 2 m.

• At 1 m: 100 / 12.566 = 7.958 W/m².
• At 2 m: 1.989 W/m² (1/4).
• At 10 m: 0.0796 W/m² (1/100).

The Sun emits about 3.828 × 10²⁶ W. At Earth's distance of 1.496 × 10¹¹ m this gives about 1361 W/m², the solar constant.

The law needs a source that is small next to the distance. Close to a long fluorescent tube or a large panel, intensity falls more slowly than 1/r².

The graph of I = k/r² is a falling curve that drops steeply near the source and flattens far away, never reaching 0.

• At r = 1 the curve is at 1; at r = 2 it is 0.25; at r = 3 it is 0.111.
• Vertical asymptote at r = 0 and horizontal asymptote at I = 0.
• The slope is −2/r³: the curve falls 3 times faster than it would under 1/r near the source.
• On a log-log graph the curve is a straight line with slope −2.

For sound from a point source, the level drops by 20 · log₁₀(d₂/d₁) decibels, about 6 dB for every doubling of distance. The sound intensity drops to 1/4 per doubling, and 10 · log₁₀(1/4) = −6.02 dB.

• 100 dB at 1 m becomes 94 dB at 2 m, 88 dB at 4 m and 82 dB at 8 m.
• 10 times the distance lowers the level by 20 dB.
• A line source, such as a busy road, loses only about 3 dB per doubling.

Indoors, reflections from walls add to the direct sound, so the level falls less than the law predicts far from the source.

Newton's law of gravity and Coulomb's law are both inverse square laws: F = G·m₁m₂/r² and F = k·q₁q₂/r². Double the distance between 2 bodies and the force drops to 1/4.

• G = 6.674 × 10⁻¹¹ N·m²/kg² and k = 8.988 × 10⁹ N·m²/C².
• A 70 kg person weighs 686.7 N at Earth's surface (1 Earth radius from the centre) and 171.7 N at 2 Earth radii.
• 2 charges of 1 μC, 1 m apart, push on each other with 0.008988 N; at 3 m the force is 0.000999 N.

The inverse square law is one line in any spreadsheet or programming language: I₂ = I₁ · (d₁/d₂)².

• Excel and Google Sheets: =A1*(B1/C1)^2 with I₁ in A1, d₁ in B1 and d₂ in C1.
• Python: I2 = I1 * (d1 / d2) ** 2.
• JavaScript: const I2 = I1 * (d1 / d2) ** 2;
• Distance: =B1*SQRT(A1/C1) in Excel, d1 * math.sqrt(I1 / I2) in Python.

The inverse square law table lists the intensity ratio I₂/I₁ = 1/(d₂/d₁)² for common distance ratios. Click a row to load it into the calculator with I₁ = 100 at d₁ = 1.

Values to know: 2× the distance gives 1/4, 3× gives 1/9, 10× gives 1/100 and half the distance gives 4×.

An inverse relation falls as 1/r, an inverse square as 1/r² and an inverse cube as 1/r³; the higher the exponent, the faster the drop with distance. At r = 2 the 3 give 0.5, 0.25 and 0.125.

Radar echoes fall as 1/r⁴: the signal spreads by 1/r² on the way out and again on the way back.

The inverse square law sets how bright a lamp looks, how loud a speaker sounds, how strong a radiation dose is and how much sunlight each planet gets.

• Photography: moving a flash from 1 m to 2 m costs 2 stops of light (1/4).
• Radiation safety: distance is a main shield; 3 times the distance gives 1/9 of the dose rate.
• Astronomy: Mars at 1.52 AU gets 1361 / 1.52² = 589 W/m², 43% of Earth's sunlight.
• Wi-Fi and radio: free-space signal strength falls as 1/r², about 6 dB per doubling.
• Plant lights: raising a grow light from 30 cm to 60 cm cuts the light on the leaves to 1/4.