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:
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.
where E is the illuminance in lux, 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 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.
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.
Luminous Flux Versus Luminous Intensity
Lighting datasheets may list either luminous flux (lumens) or luminous intensity (candelas). Flux measures the total visible light emitted in all directions, while intensity measures the light emitted in a specific direction. For a perfect point source that radiates evenly, converting from candelas to lumens is as simple as multiplying by . Real fixtures rarely behave perfectly, so you may need to consult manufacturer-provided beam distribution charts to determine how much light actually reaches your subject.
Accounting for Reflective Surfaces
Walls, ceilings, and even snow-covered ground reflect light back toward your subject, effectively boosting illuminance beyond what the simple inverse-square law predicts. Photographers working in small studios often exploit this by aiming strobes toward white ceilings to create softer, more even light. Conversely, dark matte surfaces absorb light and can decrease practical brightness. If accuracy matters, include reflectance factors in your calculations or take measurements with a lux meter after setting up the space.
Dynamic Applications
The law is essential in fields as varied as horticulture, where growers position grow lights to deliver specific photosynthetically active radiation, and workplace safety, where engineers ensure warehouse aisles meet minimum lighting standards. In astrophysics, distance measurements rely on how much brightness from a known standard candle reaches telescopes. Adjusting the calculator for these uses may involve unit conversions—for example, translating lux to foot-candles by dividing by 10.764 or converting lumens to photosynthetic photon flux density (PPFD) using spectral data.
Planning Multi-Light Setups
If you use several lights, compute each source’s contribution at the subject distance and add the lux values together. This is particularly helpful for stage designers balancing spotlights with wash lights. Keeping a spreadsheet of distances and intensities allows you to plan scenes before stepping into the venue, saving time during rehearsals.
Limitations and Advanced Models
The inverse square law assumes a point source in free space. Large fixtures, directional reflectors, or lenses can alter the rate at which light falls off. In those cases, photometric files in IES format provide detailed candela distributions. Lighting design software can import these files to simulate complex environments. Still, this calculator offers a fast approximation, especially when you need a quick feasibility check on location.
Recommended Measurement Practices
Whenever possible, validate the calculator’s output with a calibrated lux meter. Take readings at multiple distances to see how closely reality matches theory. A consistent deviation may indicate that your light has a narrower beam or that an obstruction is blocking part of the output. Documenting these readings builds a reference library that speeds up future setups.