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Inverse Square Law Calculator
See how light, sound or radiation weakens with distance. Enter a known value at one distance to get the value at another, or find how far away you need to be to reach a target level.
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Enter your values to see the result.
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How to use the Inverse Square Law Calculator
- Choose the quantity you are working with.
- Pick what to solve for: the value at a new distance or the distance needed.
- Enter the known value and its distance, then the new distance or target value.
- Read the result and the fall-off factor.
- Use the table and chart to compare other distances.
Inverse square law formulas
- Intensity at a new distance: I₂ = I₁ × (d₁ / d₂)²
- Distance for a target intensity: d₂ = d₁ × √(I₁ / I₂)
- Sound pressure level (dB): L₂ = L₁ − 20 × log₁₀(d₂ / d₁)
- From source power: I = P / (4πd²) for a source radiating evenly in all directions
Example: a desk lamp gives 800 lux at 1 m. At 2 m that becomes 800 × (1/2)² = 200 lux, and at 0.5 m it is 800 × 2² = 3,200 lux. Moving a reading light closer is far more effective than buying a brighter bulb.
Fall-off at a glance
| Distance (× d₁) | Intensity remaining | Change in dB |
|---|---|---|
| 0.5× | 400% | +6.0 dB |
| 1× | 100% | 0 dB |
| 2× | 25% | −6.0 dB |
| 3× | 11.1% | −9.5 dB |
| 4× | 6.25% | −12.0 dB |
| 5× | 4% | −14.0 dB |
| 10× | 1% | −20.0 dB |
When the law does not hold
- Extended sources: long tubes, line-array speakers and large panels fall off more slowly up close (closer to 1/d for a long line).
- Beams: lasers, spotlights and focused antennas concentrate energy, so the simple law applies only far from the source.
- Rooms and absorption: reflections indoors keep sound levels higher than predicted, while air, fog or shielding absorb extra energy.
Frequently asked questions
What is the inverse square law?
For a point source radiating evenly in all directions, intensity is inversely proportional to the square of distance: I ∝ 1/d². So I₂ = I₁ × (d₁ / d₂)².
What happens when you double the distance?
The intensity falls to a quarter (25%). Tripling the distance leaves one ninth (about 11%), and ten times the distance leaves 1%.
How does sound level in dB change with distance?
Because decibels are logarithmic, sound pressure level drops by 20·log₁₀(d₂/d₁) dB, about 6 dB for each doubling of distance from a point source in open air. 100 dB at 1 m becomes about 80 dB at 10 m.
Does the inverse square law apply to light from a lamp?
Yes, once you are several times farther away than the size of the lamp. Close to large or reflectorised fixtures, LED panels or fluorescent tubes the light falls off more slowly.
How is it used for radiation safety?
Distance is one of the three basic protections along with time and shielding. Doubling your distance from a small gamma source cuts the dose rate to a quarter. Always follow your site's radiation safety rules.
Why does it follow a square?
The energy spreads over the surface of a sphere, whose area is 4πd². Twice the radius means four times the area, so each square metre receives a quarter of the energy.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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