Display Calculations
Aspect Ratio
If you are replacing an older nonwidescreen TV with a widescreen one, one thing you probably want to know is how big of a widescreen TV you need to get to have the same 4:3 diagonal. To take an example, suppose you are replacing a 20inch (20") nonwidescreen TV.
As discussed in the Aspect Ratio article, nonwidescreen has an aspect ratio of 4:3 while widescreen has an aspect ratio of 16:9. Since 9/3=3 and 4*3=12, a 16x9 rectangle will have a central 4:3 region of 12x9. The next step is to find the length of the diagonal of the 16x9 and 12x9 rectangles.
If you take any rectangle and cut it in half along its diagonal, you get a right angle triangle. As you may recall from your days in school, the Pythagorean theorem says that if you know the length of two sides of a right triangle you can find the length of the third using the following formula:
length^{2} + height^{2} = diagonal^{2}, which is more commonly expressed as a^{2} + b^{2} = c^{2}.
Let's start with the 12x9 rectangle. As it turns out, determining the diagonal for a 4:3 rectangle is extremely simple since a 4x3 rectangle has a diagonal length of 5. So, for a 12x9 rectangle, you can simply divide 12/4 or 9/3 to get 3, which tells us that the diagonal of a 12x9 rectangle is 15 (3*5=15).
For a 16x9 rectangle, we have to use the Pythagorean theorem. This is shown below:
16x9 diagonal = sqrt(16^{2} + 9^{2}) = sqrt(256 + 81) = sqrt(337).
The rectangles below illustrate what we've worked out so far:

9 

9  
12 
16 
Now that we have put this information together, we can determine how big of a widescreen TV you need to get to have a certain sized 4:3 diagonal using the following equation:
4:3 diagonal  =  16:9 diagonal 
15  sqrt(337) 
Returning to the original example, finding a widescreen TV with a central 20" 4:3 diagonal, it becomes simply a matter of plugging in the numbers. It is worked out below:
16:9 diagonal with a 20" 4:3 diagonal = sqrt(337) * 20/15 = around 24.48".
Some older widescreen computer monitors had a 16:10 rather than 16:9 aspect ratio, which made the older widescreen computer monitors not quite as wide as they are today. Finding the central 4:3 diagonal of a 16:10 rectangle is similar to what we did above, but a little different.
First off, it should be noted that the 16:10 aspect ratio is described as 16:10 mainly for comparison purposes with 16:9. Normally when you state aspect ratio you simplify the numbers as much as you can, and 16/10 simplifies to 8/5. Therefore, it is more accurate to describe 16:10 as 8:5. The diagonal of an 8x5 grid is worked out below:
8x5 diagonal = sqrt(64 + 25) = sqrt(89).
The next issue we run into is that 3 does not go evenly into 5. Therefore, both the 8x5 and 4x3 rectangles need to be adjusted to match each other in height. 8x5 becomes 24x15, while 4x3 becomes 20x15. The diagonal of a 20x15 rectangle is 25, while the diagonal of a 24x15 rectangle is 3 * sqrt(89). Therefore, to find out the central 4:3 diagonal of an 8:5 display, you plug in numbers for the following equation:
4:3 diagonal  =  8:5 diagonal 
25  3 * sqrt(89) 
Finally, 1366x768 displays have an aspect ratio that is very close to 16:9 but slightly wider. To figure out the 4:3 equivalent for a 1366x768 display, the first thing we'll need to do is to find the aspect ratio of 1366x768, which is 683:384. Since 384/3=128 and 4*128=512, a 683x384 rectangle would have a central 4:3 region of 512x384. The 512x384 rectangle would have a diagonal of 640. The math needed to determine the diagonal of a 683x384 rectangle is shown below:
683x384 diagonal = sqrt(466,489 + 147,456) = sqrt(613,945)
This gives us the following:
4:3 diagonal  =  1366x768 diagonal 
640  sqrt(613,945) 
Pixel Density
Pixel density refers to how closely packed together the pixels are in a display. A 23" 1920x1080 display, for instance, has a higher pixel density than a 24" 1920x1080 display. The most common way to measure pixel density is to use the pixels per inch (PPI) measurement.
PPI takes the approximate number of pixels that lie along any given inchlong line on the display. This is determined by taking the diagonal of a display in pixels and dividing it by the diagonal of display in inches. To take an example, suppose we are dealing with a 50" 1920x1080 HDTV. First, we have to determine the diagonal of a 1920x1080 display in pixels. Since we know that the diagonal of a 16x9 rectangle is the square root of 337, dividing 1920 by 16 or 1080 by 9 tells us that the diagonal of a 1920x1080 display in pixels is 120 times the square root of 337. Dividing this number by 50 gives us about 44.06, meaning that a 50" 1920x1080 display has a PPI of about 44.06.
Although knowing the PPI of an HDTV may be a neat thing to know, it is not really all that useful. Knowing the PPI of a computer monitor, however, is quite important. This is because a high PPI can make fonts and icons harder to read. Windows can adjust the size of the fonts and icons to compensate for this. Not surprisingly, it is easier to properly compensate for a high PPI if you actually know the PPI of the display. Therefore, using Sven Neuhas's handdandy DPI Calculator / PPI Calculator, I constructed the table below showing the PPI of several popular size and resolution combinations for computer monitors:
Inches  Nonwidescreen  8:5 (16:10) Widescreen  16:9 Widescreen 
1515.6  15" 1024x768, 85.33  15.4" 1280x800, 98.02 15.4" 1920x1200, 147.02 
15.6" 1366x768, 100.45 15.6" 1920x1080, 141.21 
1717.3  17" 1280x1024, 96.42  17" 1440x900, 99.89 17" 1920x1200, 133.19 
17.3" 1600x900, 106.11 17.3" 1920x1080, 127.34 
18.519  19" 1280x1024, 86.27  19" 1440x900, 89.37  18.5" 1366x768, 84.71 
2020.1  20.1" 1600x1200, 99.5  20" 1680x1050, 99.06  20" 1600x900, 91.79 
21.524  NA  22" 1680x1050, 90.05 
21.5" 1920x1080, 102.46 23" 1920x1080, 95.78 24" 1920x1080, 91.79 
2730  NA  30" 2560x1600, 100.63  27" 2560x1440, 108.79 
PPI is further discussed in the Display Buying Tips article.
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