Inverse Square Law Calculator: I = P / (4 pi r^2)

How the strength of light, sound or radiation falls with distance from a small source.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • I = \frac{P}{4\pi r^2}
  • r = \sqrt{\frac{P}{4\pi I}}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the intensity: intensity = (1 / (4 x pi) x power of the source) / (distance from the source^2)
  2. Put the values in (every value in its SI unit): intensity = (1 / (4 x pi) x 100) / (2^2)
  3. The answer: intensity: 1.98944 W/m2
  4. The same answer in other units: 1989.44 mW/m2

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Distance from the source; Intensity (leave it as it is if this is what you are finding) 1.98944 W/m2; Power of the source (leave it as it is if this is what you are finding) 100 W.

  1. The formula, arranged to find the distance from the source: distance from the source = square root of ((1 / (4 x pi) x power of the source) / (intensity))
  2. Put the values in (every value in its SI unit): distance from the source = square root of ((1 / (4 x pi) x 100) / (1.98944))
  3. The answer: distance from the source: 2 m
  4. The same answer in other units: 200 cm; 0.002 km

Common questions

What is the inverse square law?

The intensity of radiation from a point source is proportional to one over the distance squared.

What happens if I double the distance?

The intensity falls to a quarter.

Does it apply to sound?

Yes, in the open air away from reflecting surfaces. Sound falls by 6 dB each time the distance doubles.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics text (the inverse square law).

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