**How to: identify geometric angles and approximate lengths to position triple-monitor setups in order to achieve a desired viewing angle**

**UPDATE 7/10/2011:**

mike.mg has compiled all of these calculations into an easy-to-use Excel spreadsheet. See his post here for the download link.

Or if you want to do the math, continue reading...

mike.mg has compiled all of these calculations into an easy-to-use Excel spreadsheet. See his post here for the download link.

Or if you want to do the math, continue reading...

*Note that this guide will probably only be useful to those who have not yet purchased their displays. Or for those who already own three displays and are simply OCD (me) to verify/reassess the geometric configuration of said monitors.*

First off, let me explain what you can identify using these calculations:

1) The length between you and your side monitors

2) The viewing angle on your side monitors

3) The angle between your two side monitors

And you can manipulate these values to determine any angle (more than likely the angle between your side monitors and your center monitor) to set up your new triple-monitor configuration.

Refer to the below photo:

In the steps below I will outline the method to identify angle x, or the angle between your two side monitors (since this is more than likely the only important detail you would want to know). However, like any mathematical formula, this can be manipulated so you can identify different values if need be.

**Step 1: Identifying your display's horizontal size**

Use the Pythagorean Thereom to identify the a and b values of the monitor given c, the diagonal size of the monitor. To determine the a:b ratio (the aspect ratio), divide the horizontal number of pixels by the vertical number of pixels. In my case it would be 1920:1080, or 16:9. This tells me there are 16 units of length for every 9 units of height.

We can determine the a and b values by first isolating a variable a:b proportion and then using substitution and plugging it in to the Pythagorean theoream in order to work with a single variable.

After determining the value of b, we can use substitution again and plug b into the Pythagorean Theorem to determine value a, the horizontal screen size.

However, since manufacturer screen size specifications do not compensate for the bezel length, we'll add 1 inch to the horizontal screen size, thus making it 21.5 inches.

**Step 2: Begin determining approximate lengths**

With the above knowledge of the horizontal screen size, we can further implement this by using the Pythagorean Theorem again to determine approximate lengths between you and your monitor. The only variable in this step is the length between you and your center monitor (assuming you are looking straight at it towards the center of your center monitor). In my case it will be 30 inches.

Here are the values we know thus far:

And since you are looking directly at the center of your center monitor, we can assume the values to the left and right of the 90 degree angle are the same. And as such we can also assume the same lengths apply to your side monitors (assuming you have three identical monitors):

We can use the Pythagorean Theorem to determine the length between you and the bezel of your two monitors, and we can further use that length to determine the length between you and the center of your side monitors.

Note that you can only determine the length between you and your center monitors if the viewing angle is assumed to be 90 degrees. If it is not, we will get to that later.

**Step 3: Determining approximate angles**

With the above knowledge we can begin using basic trigonometric formulas to determine approximate angles. We can first determine the angle between the side monitor and you. But remember since that angle is only half of the entire obtuse angle, we must multiply that value by 2.

And as such we can determine the angle between two side monitors, assuming you wanted the viewing angle to be a perfect perpendiclar angle:

**If the viewing angle between your two monitors is not perfectly perpendicular...**

Since the Pythagoream Theoream only applies to right triangles, this makes things slightly more difficult.

First, you must determine the desired viewing angle, or the angle between you and the center of your side monitor. I'm going to use 40:

And we can thus use the trigonometric cosine function to determine the length of the smaller triangle on the C₂ length (z). We can also determine value y in order to determine angle x for calculating the longer length on C₂.

EDIT 7/10/2011: An easier way to calculate C₂ is using the Pythagorean Theorem knowing the value of y (using trigonometric functions) and the length of C₁, then adding the value of z. Man, it's been a while since I've done any trig...

Thanks to mike.mg for pointing this out

Knowing these values allow us to find the angle of between the two side monitors by using more simple trigonometric functions:

And we translate these values into a view that is slightly easier to see:

It was my mistake to have chosen 40 as a degree--it is impossible to achieve that angle when the monitors are facing in, thus throwing my results off. Choosing a more realistic number would give you accurate results.

EDIT 7/10/2011: In order for all of this to work properly, you must choose an angle greater than 55 degrees.

Thanks for reading.

Edited by kiwiasian - 7/10/11 at 7:40am