inverse square law calculator

What is the inverse square law?

The inverse square law describes how intensity changes as distance from a point source increases. If you move twice as far from the source, intensity becomes one-fourth. Move three times as far, and intensity becomes one-ninth. This relationship appears across physics, including light, sound, gravity, and radiation.

The core equation

For two distances from the same source, the relationship is:

I₂ = I₁ × (d₁/d₂)²

• I₁: intensity at the reference distance d₁
• I₂: intensity at the new distance d₂
• Distance units must match (meters with meters, feet with feet)
• Intensity units can be anything consistent (lux, W/m², etc.)

Useful rearrangements

• I₁ = I₂ × (d₂/d₁)²
• d₂ = d₁ × √(I₁/I₂)
• d₁ = d₂ × √(I₂/I₁)

How to use this calculator

1. Select which variable you want to solve for.
2. Enter the other three values.
3. Click Calculate.
4. Read the result and interpretation shown below the buttons.

This tool assumes an ideal point source in free space. Real environments may include reflections, absorption, and directionality that change results.

Practical examples

1) Light intensity from a bulb

If intensity is 400 lux at 1 meter, then at 2 meters it becomes: 400 × (1/2)² = 100 lux. That is a 75% drop just by doubling distance.

2) Sound level behavior

In open space, sound intensity follows an inverse square trend. Engineers often remember this as roughly a 6 dB drop for each doubling of distance from a point source (ideal conditions).

3) Radiation safety checks

For many radiation sources, increasing distance is one of the fastest ways to reduce exposure. If you can increase distance by a factor of 3, idealized intensity drops by a factor of 9.

Common mistakes to avoid

• Mixing distance units (for example, using feet for one distance and meters for the other).
• Using negative or zero distances.
• Applying the equation in near-field conditions where source geometry dominates.
• Ignoring barriers, reflections, and atmospheric effects in real measurements.

When the inverse square law is less accurate

The model is best for point-like sources and far-field conditions. It becomes less accurate when:

• The source is large compared with distance.
• The medium absorbs or scatters strongly (fog, walls, water, tissue).
• The source is directional (spotlights, antennas, focused emitters).
• Reflections dominate (small rooms, enclosed spaces).

Quick takeaway

Distance matters more than most people expect. Because intensity scales with 1/d², small increases in distance can produce big reductions in exposure or signal strength. Use this calculator for fast planning, then validate with real measurements when precision matters.