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Uniform-Cone Lumens, Candela, Lux and Effective Area

Relate luminous flux, uniform intensity, normal illuminance, full cone angle, distance and equal-radius effective area with selectable units and inverses.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A uniform 1 cd source across a 180° full cone emits 2π ≈ 6.2832 lm. At 2 m, normal illuminance is 0.25 lx and the equivalent equal-radius spherical-cap area is 8π ≈ 25.133 m².

Ω = 2π(1 − cos(θ/2)); Φ = IΩ; E = I/r²; Aeff = Φ/E = Ωr².

Worked example

Enter these known values and leave the other values blank.

Uniform luminous intensity I
1 cd
Full cone angle θ
180 deg
Distance r
2 m
Luminous flux Φ
6.283 lm
Normal illuminance E
0.25 lx
Equal-radius effective area Aeff
25.133 m²

Assumptions and limitations

  • Intensity I is uniform inside an ideal circular cone. θ is the full apex angle, not its half angle or the solid angle Ω. The retained positive model uses 0 < θ ≤ 360°, positive intensity/flux/illuminance/distance/area and the principal acos angle inverse.
  • For a point source, E=I/r² is normal illuminance on a surface perpendicular to the arriving ray at distance r. It does not include incidence-angle cosine losses, attenuation, reflections, multiple sources or an extended luminaire. Use a photometric distribution for nonuniform real beams.
  • The area here is Φ/E, equivalent to an equal-radius spherical-cap surface Ωr² under this uniform model. It is not the footprint on a flat wall or an average wall illuminance. A flat circular beam footprint would use π(r tan(θ/2))² for the appropriate forward cone, with changing distance and incidence across that plane.
  • The initial full angle 360° represents uniform emission into 4π sr. A finite positive flux/intensity ratio above 4π is impossible in this model and is rejected. Clear an unknown before solving an inverse.
  • Lux is lm/m². One foot-candle is one lm/ft² using the exact international foot. Angle choices and compound length controls preserve physical quantities. The soccer-field area selector is the fixed 105 m × 68 m comparison (7,140 m²); actual pitches vary.
  • This tool does not infer luminous efficacy, electrical wattage, visual brightness, exposure limits, fixture counts or lighting-code compliance.

Common questions

Is the area the light patch on a flat wall?

No. It is the equivalent equal-radius spherical-cap area Φ/E=Ωr². A plane receives rays at different distances and angles, so its footprint and average illuminance require different geometry.

What is the difference between candela and lumen?

Candela describes luminous intensity per unit solid angle. Lumen is total luminous flux. Uniform intensity multiplied by the modeled solid angle gives flux.

Why can the angle inverse be impossible?

A full sphere contains 4π steradians. Positive uniform intensity cannot emit more than 4π times that intensity under this model, so a larger ratio is rejected.

References

A uniform-cone luminous flux cannot exceed 4π times its luminous intensity. The effective equal-radius illuminated area cannot exceed the area of a sphere at the stated distance. These bounds reject impossible solid angles while retaining all positive captured quantities and the inclusive 360° sphere. The inverse cone angle is evaluated only for positive intensity and 0 < flux ≤ 4π intensity, so an impossible acos argument leaves angle blank with a physical-domain error rather than consuming known inputs through a complex result. Foot-candles use the exact international square foot rather than the rounded captured illuminance factor. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.

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