The Inverse Square Law

Light spreads out as it travels away from its source. Imagine an imaginary sphere centered on a bare bulb. As the sphere grows, its surface area increases in proportion to the square of its radius. Because the same amount of light energy must cover a greater area, the intensity at any given point on the sphere diminishes rapidly with distance. This concept, known as the inverse square law, can be written as:

where E is the illuminance in lux, \(\Phi\) represents the total luminous flux in lumens, and d is the distance in meters. The equation assumes the source radiates uniformly in all directions, like an idealized point light. A real light fixture may distribute brightness unevenly, but the inverse square approximation still offers a remarkably good starting point for planning photography lighting, stage design, or outdoor illumination.

The calculator above accepts luminous flux in lumens because that quantity is often listed on consumer lightbulb packaging. If you instead know the luminous intensity in candelas, you can multiply it by \(4\pi\) to obtain lumens for an isotropic emitter. Either way, once you input the total flux and your distance from the source, the tool returns the predicted illuminance in lux.

The table below illustrates the effect of doubling or tripling distance for a 1000 lumen source. Notice how drastically illuminance falls as you move away:

Distance (m)	Illuminance (lux)
1	79.6
2	19.9
3	8.8
4	5.0

Such sharp decline can be challenging when designing lighting for photographs or architectural spaces. Moving a light just a few meters farther away might cut its brightness to a quarter of what it was before. Conversely, bringing a light closer dramatically increases intensity, but at the expense of uniform coverage across the subject.

In astrophysics the same law explains why stars that are extremely luminous may still appear faint if they are far from Earth. Measuring brightness from a known distance allows scientists to estimate the true power of cosmic phenomena. For everyday purposes, photographers use the rule when deciding how much flash power they need for a portrait. Stage technicians determine spotlight positions so that actors remain visible under bright but not overpowering lights. Homeowners use it to size floodlights for gardens and security.

Although lux is an SI unit meaning lumens per square meter, older literature sometimes discusses footcandles. If you need footcandles, simply divide lux by 10.764. Likewise, you can back-calculate luminous flux if you know your desired lux level and the distance from the source. Algebraically, \(\Phi=4\pi d^2 E\).

Keep in mind that reflectors, lenses, or diffusers attached to a bulb can change its emission pattern, causing the simple equation to deviate. Nevertheless, the inverse square model is invaluable for first-pass calculations. It helps avoid over- or under-lighting a scene, saving time and energy. In the table above we see that a 1000 lumen bulb only produces around 5 lux at four meters—a surprisingly small amount. To brightly illuminate a driveway or flagpole from that distance, you might need multiple fixtures or a purpose-built spotlight.

Some lighting designers employ the concept of “throw distance” to indicate how far a fixture can deliver adequate illumination. This calculator effectively tells you the expected illuminance at any given throw. If you need to maintain 200 lux for reading or 50 lux for casual outdoor gatherings, you can plug in distances until the formula shows the required brightness.

Experiment with different values to see how orientation, reflector efficiency, or diffusion material might influence your setup. Remember also that our eyes adjust to a wide range of lighting conditions. While 100 lux may feel dim for working on small parts, 5 lux might be perfect for accent lighting along a walkway. By understanding the inverse square relationship, you gain more control over these decisions.