# Projector Math

## Disclosure & Overview

<span style="background-color: rgb(0, 0, 0); white-space: pre-wrap;">In all the eexamples below, using a 1920x1080 projector at 8’ away projecting 16’ wide x 9’ high. </span>

<span style="white-space: pre-wrap;">Special shout out here to Sean Leo who hooked me up with some of these. </span>

<span style="white-space: pre-wrap;">You need to use projector math for all kinds of things – but mostly for </span>[Projector Positioning](http://tech-almanac.org/books/displays/page/projector-positioning "Projector Positioning")<span style="white-space: pre-wrap;"> and engineering. </span>

## Throw Ratio Formulas

To get Projector Width = Distance x Throw Ratio

8 x .5 = 16

To get Projector Throw Ratio = Distance / Projector Width

8 / 16 = .5:1

To get Projector Distance = Throw Ratio x Width

.5 x 8 = 16

## Diagonal Dimensions

<span style="background-color: rgb(0, 0, 0);">// Based on a single Projector in a blend not the blended resolution</span>

****W****<span style="white-space: pre-wrap;"> Width = 16 ft to inches x 12 = 192 ″</span>

****H**** Height = 9 ft to inches x 12 = 108 ″

****pxW****<span style="white-space: pre-wrap;"> Pixel Width = 1920//pixel Width = 1920</span>

****pxH****<span style="white-space: pre-wrap;"> Pixel Height = 1080//pixel Height = 1080</span>

****inW****<span style="white-space: pre-wrap;"> Inches in Width = W//inches Width = 192”</span>

****inH****<span style="white-space: pre-wrap;"> Inches in Height = H//inches Height = 108”</span>

<span style="background-color: rgb(0, 0, 0);">//functions</span>

****pxD****<span style="white-space: pre-wrap;"> Diagonal Pixels = sqrt (</span>****pxW****<span style="white-space: pre-wrap;">^2 + </span>****pxH****^2) = 2202.90717

****inD****<span style="white-space: pre-wrap;"> Diagonal Inches = sqrt (</span>****inW****<span style="white-space: pre-wrap;">^2 + </span>****inH****^2) = 220.290717

## Pixels Per Inch (PPI)

<span style="white-space: pre-wrap;">There are too many ways to calculate PPi. In an array of projectors, you should use values from a single projector, as the values from an array of projectors will give you different PPi for Height than Width (even though they are the same if the projectors are spec’d the same). I don’t really understand why it happens this way. </span>

****pxD****<span style="white-space: pre-wrap;"> Pixel Diagonal / </span>****inD****<span style="white-space: pre-wrap;"> Inch Diagonal of Screen = 10</span>

## Aspect Ratio

<span style="white-space: pre-wrap;">The basic “Aspect” is easily calculated. It’s the </span>****pxW****<span style="white-space: pre-wrap;"> / </span>****pxH****: 1920/1080 = 1.7777777778

<span style="white-space: pre-wrap;">To get the ratio, that magical little 16:9, you need to find the GCD (greatest common divisor) of </span>****pxW****<span style="white-space: pre-wrap;"> and </span>****pxH****<span style="white-space: pre-wrap;">. You can do this by using the </span>****GCD****<span style="white-space: pre-wrap;"> function in a spreadsheet: GCD(1920,1080) will return a GCD of 120. To get the aspect, you divide the </span>****pxW****<span style="white-space: pre-wrap;"> by that GCD: 1920/120 will return 16. To get your second number, you divide the </span>****pxH****<span style="white-space: pre-wrap;"> by that same GCD: 1080/120 will return 9. </span>

<span style="background-color: rgb(0, 0, 0);">// Spreadsheet formula - replace pxH, pxW with the cells that contain those numbers</span>

=SUM(pxW/GCD(pxW,pxH))&amp;":"&amp;SUM(pxH/GCD(pxW,pxH))

## Lumens

Foot Lamberts (fL) = (Projector Brightness in Lumens / Area of Screen in SqFt) \* Gain