Beam angles, spot sizes and the inverse square law

The three pieces of hand arithmetic that let you sketch a scheme in a meeting without opening Dialux.

Ba phép tính nhẩm giúp bạn phác một phương án ngay giữa cuộc họp mà không cần mở Dialux.

Calculation software is for verification. In a design meeting you need answers in seconds, and three formulas cover most of what gets asked.

Inverse square law

Illuminance from a point source falls with the square of distance:

E = I / d²

With the surface tilted at angle θ from the beam’s normal:

E = I × cos θ / d²

So a 5000 cd spotlight at 4 m gives 5000 / 16 = 313 lux on axis. At 6 m it gives 139 lux. Moving a fixture 50% further away costs you 56% of the light.

The validity limit matters. The inverse square law assumes a point source, which in practice means the distance must be at least about five times the largest dimension of the emitting surface. It is fine for a downlight at 3 m. It is wrong for a 1.5 m linear fitting at 2 m, where the falloff is closer to 1/d.

Beam angle to spot size

Beam angle is the full angle between the two directions where intensity has fallen to 50% of peak. Field angle is the same idea at 10%, and describes the visible spill.

spot diameter = 2 × distance × tan(beam angle ÷ 2)

A 36° beam at 3.25 m throw lands a 2.11 m pool. Beam angle is measured where intensity falls to 50% of peak; the paler field angle, at 10%, is the spill you also have to design for.

At 4 m throw:

Beam	Spot diameter	Character
10°	0.7 m	Object accent, single sculpture
15°	1.05 m	Artwork, table centre
24°	1.70 m	General accent, seating group
36°	2.60 m	Wide accent, circulation
60°	4.60 m	General downlight

Two numbers worth memorising: at a 3 m throw a 24° beam gives roughly a 1.3 m pool, and a 60° beam roughly 3.5 m. Most residential and hospitality sketching happens around those.

Spacing to height ratio

For even general lighting on a horizontal plane, spacing between luminaires should not exceed roughly 1.0–1.5 × the mounting height above the working plane, depending on the distribution.

For a 2.7 m ceiling and a 0.8 m desk, the height above the plane is 1.9 m, so downlights want to sit at roughly 1.9–2.6 m centres. If the architect’s reflected ceiling plan shows 4 m centres, the room will be scalloped and patchy no matter which fixture you pick.

Wall washing geometry

To wash a wall evenly rather than graze it, the rule of thumb is:

Offset from the wall ≈ one third of the wall height
Spacing between washers ≈ the same as the offset, or slightly more

For a 3 m wall: fixtures roughly 1 m off the wall at roughly 1–1.2 m centres. Move to 0.15 m off the wall and you are no longer washing, you are grazing — which is a legitimate choice, but a different one, and it will reveal every imperfection in the plaster.

Same wall, same luminaire, two offsets. Washing sits roughly a third of the wall height away and flattens the surface into an even plane. Grazing sits close and rakes the light across it, so every texture throws a shadow. Neither is more correct — they answer different questions about the material.

Working example

A 4 m high hotel lobby. The client wants a 2 m diameter pool of light on a round table, at roughly 300 lux, with a sharp edge.

Throw is 4 m minus 0.75 m table height = 3.25 m.
Required beam ≈ 2 × arctan(1.0 / 3.25) = 34°. Specify a 36° optic.
Required peak intensity: I = E × d² = 300 × 3.25² = 3170 cd.
Ask the manufacturer for a 36° fitting with ≥3200 cd centre-beam. If the datasheet says 2000 cd, you need two fittings, a narrower optic, or a lower target.

That is a complete, defensible answer in about ninety seconds, and it is right to within the tolerance of the finishes.

Do not skip the software

Hand calculations get you the concept and keep you credible in a room. They cannot handle interreflection, and interreflection is where 20–40% of the light in a bright, light-coloured room comes from. Sketch by hand, verify in Dialux, Relux or AGi32, and always mock up anything where texture or a specific finish is doing the work.