Methods to Determine Fixture Angles, Throw
Distances, Light Pool Sizes, and Light Levels
as Correlate to Any Stage Lighting Fixture
THE FOLLOWING MAY NOT BE REPRODUCED
WITHOUT PERMISSION FROM THE AUTHOR ©
Stage lighting fixture models provide different light angles and
intensities. Knowing ahead of time which versions to rent for
your production will save trial and error testing. This article
offers information that will enable one to confidently order
the correct fixture for the required beam size at a given
distance, and to grasp the intensity/distance concept.
For those with fixtures, the instructions provided here
will determine, without physical experimentation tedium,
the Hang Point, and/or Light Pool size, and/or Fixture Angle,
and/or Light Level. As an additional aid, some initial calculations
have been done for you so as to reduce them to one factor. This
will be used instead, thus eliminating the total number of computations.
The Why


Table of Definitions 
The Factors 
Factor Chart 
Light Pool Formula and Examples


Formula Summary 
Precision

Calculating Intensity 
Intensity Formulas and Examples

Why take time to do the mathematics that will be presented
here in this article? Expanding upon what was said in the
opening statements will answer this question:
Realise: Hanging Fixtures "on paper" will save much work
later on. The lesson below will explain how to do this.
Throughout this section and the instructions that follow, the
most important terms will be shown as firstletter capitalised.
Synonymously, firstletter capitalised terms will be defined here.
Italicised terms are defined elsewhere in this section.

After the preliminary discussion below, a chart will display a
multiplication/division Factor corresponding to a Fixture's Field
Angle. We have made the effort to reduce the computations
down to a single number to save you time and effort, and to
simplfy the process. Use this number to directly determine:
(1) The dimension(s) of the area covered by the light emitted
by a Fixture with a specific Field Angle at a specific Distance.
That is: What size will the Light Pool be
with this Fixture that far away?
or to determine:
(2) The Throw Distance needed to create a Light Pool of a
specific dimension using the Field Angle of a specific Fixture.
That is: How far away must this Fixture
be to make that size Light Pool?
or to determine:
(3) The Fixture needed to create a Light Pool of a
specific dimension at a specific Distance.
That is: Which Fixture will give me a
Light Pool of this size at that Distance?
These Chart Factors are very accurate for Fixtures that produce an even amount of light and are shining directly onto a flat surface, or pointed straight down to a floor. If the Fixture is tilted or panned, the light will spread out making a larger (and also dimmer) pool of light; so the results of using these Factors become less accurate as a Fixture tilts or pans away from straight on. They are also less accurate for Fixtures that project an uneven light.
Even when accurate though, other criteria come into play such as illdefined Light Pool edges, and spill coming from the Fixture itself. Another is that some Light Pools will be wider than desired because the actual edges fall outside the 10% threshold used by manufacturers when determining the Field Angle specification. The latter is most often seen with the PAR lamp and fresnel Fixture. The perimeter of light that falls below 10% becomes most noticeable when only one Fixture is used, and it's classed as "spill" when it is unwanted. (Further discussion is in "Precision Caveats", father on.)
Multiplication/Division Factors

ANGLE: 5° 10° 15° 20° 25° FACTOR: .09 .18 .27 .36 .45 ANGLE: 30° 35° 40° 45° 50° FACTOR: .54 .63 .72 .81 .90 ANGLE: 55° 60° 65° 70° 75° FACTOR: .99 1.08 1.17 1.26 1.35 ANGLE: 80° 85° 90° 95° 100° FACTOR: 1.44 1.53 1.62 1.71 1.80 
Directly Proportional: Studying the Chart above will show that the relationships among the Factors are in direct proportion. So if one doubles the Fixture Angle, the Factor also doubles, and vice versa. This also means that for the same Throw Distance, the Light Pool doubles in dimension(s) every time the Fixture Angle doubles. In addition, with a given Fixture Angle, doubling the Throw Distance results in a Light Pool that also doubles in dimension(s). Options: So if one needs to illuminate a space with twice the dimension(s), one can either double the Throw Distance by moving the same Fixture farther away, or keep the Fixture's position the same, but exchange it for one rated at double the angle. Lower Light Levels: Related to the above, it should be realised that when doubling the Light Pool dimension(s), the area covered will quadruple. As an example, 2 X 2 metres encompasses 4 square metres; doubling this to 4 X 4 metres will then encompass 16 square metres. Four times the area now covered means that the light intensity will be reduced to 1/4. This is because the same amount of light projected by a given Fixture will be spread out to cover four times the space; that is, it will cover 16 squares instead of 4. 
Light Pool Diameter: To determine the diameter of a Light Pool at a specific Distance and Fixture angle, the formula is:
Distance times the Factor = Light Pool DiameterSo, to find the diameter of the resulting Light Pool when shining a 25degree ellipsoidal straight on to a flat wall from a distance of 5 metres:
5 X .45 = 2.25 metresThe Light Pool in this instance will be 2.25 metres wide. This figure is derived by using the Factor taken from the Chart for a Fixture with a 25degree angle. The Throw Distance has been multiplied by this Factor to get the Light Pool dimension.
To find out the approximate dimensions of an oval projected by a PAR 64 `FFR' lamp at the same distance, one must use two Factors because the PAR lamp light output is not round. From the manufacturer's Field Angle specifications for this lamp (21 X 44 Degrees), and assuming that the barrel of the PAR Fixture does not compromise the light emanating from it by cutting off part of that light:
5 X .36 = 1.80 metres
This time, the Factors taken from the Chart are for the Field Angles
closest to the FFR lamp's specified angles of 21 and 44 degrees. Thus,
the Chart Factors associated with `20' and `45' degrees were used to get
the approximate oval dimensions projected by an FFR lamp at a 5metre
Throw Distance.
Throw Distance: At what point will a 25degree Fixture project a 2.25metre Light Pool Diameter? The formula is:
Light Pool Diameter divided by the Factor = Distance
Here, the desired Light Pool dimension is divided by the Factor for a
25degree Fixture. This is to calculate at what Throw Distance the Fixture
will need to be positioned to achieve that 2.25metre Light Pool.
Fixture Field Angle: What happens if you know the Throw Distance and Light Pool dimensions, and want to know which Fixture to employ? Using the same 2.25metre diameter Light Pool projected from the same fivemetre Distance, as in the first example, this formula will answer the question:
Light Pool Diameter divided by Distance = Factor
Referring to the Chart shows that Factor `.45' corresponds to a
25degree Fixture.

Now, it may be important for some persons to have accuracy for the Fixtures in their inventories that have angles which lie between the Fixture Angle numbers shown on the Chart. One can calculate the Factors for such Fixtures by doing the following:
Fixture Degrees X .018
Thus, for a Fixture with a manufacturer's stated
angle of 37 degrees, the Factor will be:
Notice that this result falls between the Factors shown on the Chart
for 35 and 40degree Fixtures, as it should. Factors for Fixture Field
Angles not represented on the Chart could be calculated and then added,
but a better method is to print out only the Factors for Fixtures in
one's own inventory. This could be posted wherever it might be needed.
A further improvement would be to also post Light Pool Diameters for
each Fixture at a range of Throw Distances typical for that Fixture.
This would eliminate the calculation step each time.
Precision Caveats
Edge Limitations Realise that some Fixtures limit the edge
of the Light Pool to an intensity that is higher than the actual 10% Field
Angle intensity of that Fixture. Thus, that actual Field Angle intensity is
not available. However, the angle stated by the manufacturer will still work
for all calculations shown in this article because this angle defines the
maximum edge of its Light Pool.
Conversely, Fixtures that don't limit their light output diameter may produce Light Pools which edge intensities are less than 10% of the central one. This could be of some concern regarding manufacturers that use the actual Field Angle specification of 10% to represent the edge diameter as opposed to stating an angle that represents its Absolute Diameter. In this case, Light Pool Diameters may be slightly larger than calculated. However, outside that calculated diameter, light intensity is usually so low and levels fall off so quickly, that it won't cause concern for most lighting purposes. If it does, choose a different Fixture for your purpose, or limit the Light Pool Diameter by using internal shutters, or accessories such as barndoors, a snoot, or a funnel.
NonPerpendicular Angles As mentioned earlier, this lesson discusses setups regarding Fixtures that are shining directly at a surface. The results of the calculations will always be suitable for this purpose, and in a general context will be suitable for those Fixtures not perpendicular to a surface, unless the angles are very acute. The results of acute angles can be worked out using trigonometric functions, but this aspect is currently outside the scope of this webpage.
If it is important to know how bright a Light Pool is at a
specific Distance with a specific Fixture, this section will assist.
First, be aware of the following points that will affect the
final figure of your calculation or give varying results:
 Unless using a light meter to find an accurate measurement,
your intensity calculation will be based on the fixture manufacturer's
specification as found in a catalogue or on a website.
 Fixtures shone at an angle other than perpendicular to a surface
will have a Light Pool that is keystoned. Points in the stretched area
that are farther from the Fixture will have a lower intensity.
 Fixtures shone onto uneven surfaces or objects will have a
range of intensities that differ over the surface or object.
 Fixtures with uneven beam characteristics will not give the
same intensity over the entire Light Pool. Some areas may be
brighter or dimmer than your calculated figure.
 Fixtures that have alignment adjustments will result in an
intensity figure that is dependent on that alignment.
 Fixtures that have not been serviced for some time will produce
a lower intensity figure. This is due to dirt, deterioration, and
possible misalignment.
To determine how bright a Light Pool will be at a specific Distance with your chosen Fixture, the formula is:
Fixture Light Output divided by the Distance Squared = IntensitySo, to calculate an intensity, consult the manufacturer's specification for the light output of the Fixture you have chosen. It will be in candela, or for older Fixtures, in candle power. For this example we will assume a 1000watt ellipsoidal with a Flat Field alignment and a manufacturer's rating of 100,000 Candela. It is hung at a 10metre distance.
100,000 ÷ 100 = 1000 Lux
The Intensity in this instance will be 1000 Lux. This figure is derived
by using the manufacturer's specification for an ellipsoidal with a specific
Field Angle, lamp and alignment, which in this example is 100,000 Candela.
This number is divided by the distance of 10 metres that has been squared
to be 100.
Lumen Output
For those that may be thinking that the Fixture light output specification
should be in Lumens, this term is only applicable to light radiating in all
directions. Such a specification is used to rate the light output of the lamp
filament itself. It is not useful for the purpose intended here where light
output is concentrated by a reflector, or by a reflector and one or more
lenses, as is done in stage lighting Fixtures.
Regardless, you may wish to calculate the intensity of a single lamp in a fixture, such as one in a socket at the top of a stand. To find the light level at a given distance by such a setup, use this formula:
Lumens divided by the Distance Squared = IntensityThus a bare lamp that is placed on top of that stand, and has an output of 15,000 Lumens, will produce a light level at 10 metres of 150 Lux (15,000 ÷ 100 = 150).
Although Candela and Lumens directly equate in the above calculations, manufacturers differentiate the terms so that one will know that a "Candela" specification means that the light output has been altered through the usage of reflectors and lenses, while the "Lumen" specification has not.
Having said that, there are some projection lamps that have internal reflectors; despite this, some manufacturers rate some of these outputs in lumens. These lamps are rarely seen in stage lighting circumstances so they can typically be discounted.
Excluding Intensity, the methods discussed in this article will work
with all stage lighting fixtures regardless of type, manufacturer
or model  as long as the Field Angle of the Fixture is known.
Intensity calculations will require the manufacturer's
specification for the given fixture.
Using all the knowledge gained here will eliminate trial and error
when ordering, hanging, or designing with, any lighting instrument.
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