# AIEL Instructional STAGE LIGHTING FIXTURE CALCULATIONS

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 ©

## PRELIMINARY

Stage lighting fixture models provide different light angles and
intensities. Knowing ahead of time which versions to rent for
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.

The instructions provided here will determine, without
physical experimentation tedium, the Hang Point, and/or
Light Pool size, and/or Fixture Angle, and/or the Light Level.
As an additional aid, some initial calculations have been done
for you that allows them to be reduced to a single factor. This will
be used instead, thus eliminating the total number of computations.

## Topics

### The WHY

Why take time to do the mathematics that will be presented
opening statements will answer this question:

perform in a given situation means not having to experiment
with hanging and rehanging Fixtures; you will already
understand where to place them and why.

2.   If you must buy or rent lights for your location, discerning
which ones to order ahead of time means no disappointments
after you actually place those Fixtures.

3.   When projecting patterns, the Fixtures you have chosen will
provide exactly the desired image sizes at the selected distances
because you did the calculations before making those choices.

4.   By using the information gained from this discussion, coupled
with the manufacturers' intensity specifications, you will know
distance and Field Angle.

Realise: Hanging your plot's fixtures "on paper" will
save much work later on during the setup and test
period. The information and instructions in this
article will explain how to do this.

### DEFINITIONS

Throughout this section and the instructions that follow, the
most important terms will be shown as first-letter capitalised.
Synonymously, first-letter capitalised terms will be defined here.
Italicised terms are defined elsewhere in this section.

### The FACTORS

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 ill-defined 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 for Stage Lighting Fixtures

Use Field Angles
for Calculations

```  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:  Regarding 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    FORMULA     and EXAMPLES

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 Diameter

So, to find the diameter of the resulting Light Pool when shining a 25-degree ellipsoidal straight on to a flat wall from a distance of 5 metres:

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 25-degree 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
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 5-metre Throw Distance.

Throw Distance: At what point will a 25-degree Fixture project a 2.25-metre Light Pool Diameter? The formula is:

Light Pool Diameter divided by the Factor = Distance

so...

2.25 ÷ .45 = 5 metres

Here, the desired Light Pool dimension is divided by the Factor for a 25-degree Fixture. This is to calculate at what Throw Distance the Fixture will need to be positioned to achieve that 2.25-metre 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 2.25-metre diameter Light Pool projected from the same five-metre Distance, as in the first example, this formula will answer the question:

Light Pool Diameter divided by Distance = Factor

so...

2.25 ÷ 5 = .45

Referring to the Chart shows that Factor `.45' corresponds to a 25-degree Fixture.

## FORMULA SUMMARY To determine Throw Distance, divide Light Pool Diameter by the Factor. Doubling the Throw Distance will double the Light Pool Diameter. Doubling the Throw Distance will drop the light level to 1/4. To determine Fixture Angle, divide Light Pool Diameter by the Distance.
(Look for the resulting Factor on the Chart to get the Fixture Angle) Doubling the Fixture Angle will double the Light Pool Diameter. Doubling the Fixture Angle will drop the light level to 1/4. To determine Light Pool Diameter, multiply Distance by the Factor. Doubling the Light Pool Diameter will quadruple the area covered. Doubling the Light Pool Diameter will drop the light level to 1/4.

Remember that Panning or Tilting the Fixture
will affect the calculations because the Light
Pool will be Keystoned (stretched) to one side.

### SHORTCUT TIPS

• Tip 1:   Notice back in the Chart that for a Fixture with a 55-degree Field Angle, the Factor is almost exactly `1'. Whenever you use such a Fixture, realise that the Throw Distance and Light Pool Size will be extremely close to being equal, that is, having a Factor of `1'.(*) Thus, at a Distance of 5 metres, the Pool Size will be almost 5 metres. This relationship is true for any Throw Distance, although practicality dictates that for great Distances the light intensity will be too low to be useable.

(*)   For Throw Distance and Light Pool Size to be exactly equal, the Fixture angle would have to be 55.55... degrees. This figure is derived from dividing 1 by .018. The latter number here represents the mathematical relationship between angle and the resulting diameter of a Pool of Light.

Remember this 55-degree number. Any Fixture that has an angle greater than 55.55... degrees, will give a Pool Diameter wider than the Throw Distance; one with a lesser angle will give a Pool Diameter narrower than the Throw Distance.

• Tip 2:   For fast calculations when using Fixtures with the
following Beam Angles, realise these rough relationships:

• A 15-degree Fixture gives a Light Pool Diameter
approximately 1/4 that of its Throw Distance.

• A 20-degree Fixture gives a Light Pool Diameter
approximately 1/3 that of its Throw Distance.

• A 25-degree Fixture gives a Light Pool Diameter
approximately 1/2 that of its Throw Distance.

• A 35-degree Fixture gives a Light Pool Diameter
approximately 2/3 that of its Throw Distance.

• A 40-degree Fixture gives a Light Pool Diameter
approximately 3/4 that of its Throw Distance.

### CALCULATION PRECISION

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:

37 X .018 = .666 factor

Notice that this result falls between the Factors shown on the Chart for 35- and 40-degree 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.

Non-Perpendicular 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.

### FIGURING INTENSITY

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:

1.   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.

2.   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.

3.   Fixtures shone onto uneven surfaces or objects will have a
range of intensities that differ over the surface or object.

4.   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.

5.   Fixtures that have alignment adjustments will result in an
intensity figure that is dependent on that alignment.

6.   Fixtures that have not been serviced for some time will produce
a lower intensity figure. This is due to dirt, deterioration, and
possible misalignment.

### INTENSITY    FORMULAS     and EXAMPLES

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 = Intensity

So, 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 1000-watt ellipsoidal with a Flat Field alignment and a manufacturer's rating of 100,000 Candela. It is hung at a 10-metre 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 = Intensity

Thus 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.

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.