Illuminance at a Distance Calculator (Inverse Square & Cosine Law)

How it works

E = I ÷ d² × cos θ (E lux, I candela, d metres); I ≈ lumens ÷ [2π(1 − cos(beam/2))]; d for a target = √(I cos θ ÷ E); 1 fc = 10.764 lux

Light from a small source spreads over a sphere whose area grows with the square of the distance, so illuminance falls with 1/d²: a lamp giving 200 lux at one metre gives 50 at two and 8 at five. When the surface is tilted away from the source the same light is smeared over a larger area, reducing illuminance by the cosine of the angle — the reason winter sun is weak and the far end of a table under a pendant is dim. Manufacturers rate directional lamps in candela (intensity) or by lumens and beam angle; converting lumens to candela through the beam's solid angle gives a fair centre-beam estimate for spots and floods. The law holds for sources small compared with the distance (rule of thumb: five times the source size); long fluorescent tubes and large panels fall off more slowly up close.

Worked example

A 2,000 cd spotlight gives 2,000 ÷ 3² = 222 lux (20.6 fc) on a surface 3 m away square-on, 111 lux if the surface is tilted 60°; to get 100 lux it can be 4.47 m away. A 1,500-lumen 40° flood is about 3,970 cd and would give 441 lux at 3 m over a 2.2 m circle.

Frequently asked questions