Point Illuminance Formulas

Point-by-point lighting formulas: the illuminance on a horizontal plane from a luminous intensity, the height and the angle, and the intensity needed.

The formulas

I is the luminous intensity of the source in candela in the direction of the point, d the distance from the source to the point, h the mounting height above the plane and θ the angle between the vertical and the direction of the point. The first formula is the inverse square law and the cos θ term takes account of the tilt of the surface. The formula is for a point source. A lamp has a different intensity at each angle, so take I(θ) from the photometric data of the fitting, and an extended source close to the plane needs a full calculation. The values here are examples, and the light of other fittings and of reflections adds to the result.

Worked example: a source of 5,000 cd at 10 m

The intensity is the same in all directions in this example.

1. Directly below, E = 5000 / 10² = 50 lx
2. At 30°, E = 5000 × cos³30° / 10² = 32.48 lx, at a horizontal distance of 10 × tan 30° = 5.774 m
3. At 45°, E = 5000 × cos³45° / 10² = 17.68 lx

Worked example: the intensity for 20 lx directly below at 8 m

1. I = 20 × 8² / cos³0° = 1,280 cd

Illuminance by angle for 5,000 cd at 10 m

Frequently asked questions

Why does the illuminance fall away from the point below?

The distance to the source grows, which lowers the light by the inverse square law, and the surface is tilted to the ray, which lowers it again. Together they give the cos³θ term.

Why use the photometric data of the fitting?

A real fitting does not give the same intensity in all directions. The data gives the intensity at each angle, and the formula then gives the light at each point.

How do I find the average illuminance?

Calculate the points over a grid and take the mean, or use the lumen method in the lighting article with the utilisation factor.

Does this apply to airfield lights?

It applies to a point source of light in general. Airfield lights have standards that fix the intensity at each angle, so check the standard of your authority.

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