Ways to Determine Fixture Angles, Throw
Distances, and Light Pool Diameters as
Correlate to Any Stage Lighting Fixture
THE FOLLOWING MAY NOT BE REPRODUCED
WITHOUT PERMISSION FROM THE AUTHOR ©
Stage lighting fixture models provide different beam 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 having fixtures, the instructions provided here will
determine, without the trouble of physical experimentation,
the right hang point to attain the required Light Pool size.
The Why


Definitions 
The Factors 
Factor Chart 
Examples

Summary

Precision

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.

After the preliminary discussion below, a chart will display the
multiplication/division Factor corresponding to a Fixture's Field
Angle. Use this number to calculate:
(1) The dimension(s) of the area covered by the light emitted
by a Fixture with a given Field Angle at a given distance.
That is: What size will the Light Pool be
with this Fixture that far away?
or to calculate:
(2) The Throw Distance needed to create a pool of light of a
specified dimension using the Field Angle of a given Fixture.
That is: How far away must this Fixture
be to make that size Light Pool?
These Chart Factors are very accurate for Fixtures that produce an even amount of light and are shining directly on to 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. 
Light Pool Diameter: To find what the diameter of the light
projected will be at a distance of 5 metres when shining a 25degree
ellipsoidal straight on to a flat surface:
5 X .45 = 2.25 metres
The 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 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
5 X .81 = 4.05 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?
2.25 ÷ .45 = 5 metres
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:
2.25 ÷ 5 = .45 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 36 degrees, the Factor will be:
36 X .018 = .648 factor
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 given 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 generally 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.
The methods discussed in this article work with all stage lighting
Fixtures regardless of type, manufacturer or model  as long as
the Field Angle of the Fixture is known. Using the knowledge
gained here will eliminate trial and error when ordering,
hanging, or designing with, any lighting instrument.
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