Look up some of your possible monitors specifications here:

http://support.dell.com/support/systemsinfo/documentation.aspx?c=us&cs=19&l=en&s=dhs&~cat=3

Here is an example: 2005fp 20 inch diagonal 17 by 10 inches

http://support.dell.com/support/edocs/monitors/2005FPW/English/about.htm#Specifioications

Now you have the numbers you need, no trig!!

Well, I was hoping for a mathematical solution...

All I could figure out is that a widescreen has about only 93% of the real estate of an equivalent diagonal measure conventional screen (based on comparison of Dell's 20.1" monitors).

And at this size the widescreen is only 0.9 inches wider (and 1.4 inches shorter).

Anyone been to class lately? And I thought I'd never have a reason to use any of the math stuff in real life...

Maybe you could use cross multiplying ratios 4 width 3 height=Desired width/height for 4:3 screen. If Desired width =40 , 4/3 = DW/H Then use the width and height to get screen area, a2=b2=c2 for c the diagonal.

4/3=40/30 (40*40)+(30*30)=c2 1600+900=2500 c2=2500 c=square root of 2500=50

Something like that might work for you, no guarantees, I was Never good at math, but triangles weren't too bad.

(1)
Given monitor diagonal "X", how to I determine the width (and height) knowing just that and the aspect ratio (4x3, 5x4, or 16x9) - to allow comparison of relative monitor size (physical display area).

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You have monitor with diagonal length C.
The ratio of the monitor is I x J.

First, think of his as a right triangle with base I and height J.

|\
| \
J | \
| @\
------
I

Now, we want to calculate angle where the @ is.
To do that, we use ArcTan = (J/I) [Using a calculator]

Now we have angle @.
To calculate screen dimensions for diagonal C.

Screen Width: Width = Cos(@) * C
Screen Height: Height = Sin(@) * C

Cool? For example, you have a 17" diagonal on a 4x3 monitor.

ArcTan = (3/4)
So our angle equals 36.87 degrees.

Width = Cos(36.87) * (17") = 13.6"
Height = Sin(36.87) * (17") = 10.0"


To solve ones below you can work backwards. I will do them using the numbers you gave.
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(2)
For example... if I decide I like the width of a given 16x9 display of diagonal 24", and want to maintain that width in a 4x3 display; what size diagonal should the new monitor have?

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Ok so with a 16:9 the angle formed is ArcTan(9/16) = 29.36 degrees
And the Width will be Cos(29.36) * (24") = 20.92"

Now, remember that the angle formed on a 4x3 is different.
The angle we calculated for the 4x3 above will be same for all monitors in that ratio.

So, we know angle and a width.

To get the diagonal we do the following calculation:

Diagonal = [20.92 / Cos(36.87)]
That is the Width desired divied by the cosine of the angle from your desired ratio.

In this case a 4x3 monitor with diagonal 26.15" will have the same width as a 16x9 monitor with diagonal 24".

But, if you know angle and Height, do height of desired divided by sine of the angle from desired ratio.

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(3)
Or, what is the relative area of a 19" 5x4 display to a 19" 16x9? I can assume the 4x3 has more area, but how much more?

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To get area its just width * height.

So for a 19" 5x4: Width = 14.84" Height = 11.87" and Area = (14.84) * (11.87) = 176.15 square inches

And for a 19" 16x9: Width = 16.56" Height = 9.31" and Area = (16.56) * (9.31) = 154.17 square inches.

And for a 19" 4x3: Width = 15.2" Height = 11.4" and Area = (15.2) * (11.4) = 173.28 square inches

So for 3 monitors with same diagonal but differing ratios. the largest screen area to least goes in the following order:

5x4
4x3
16x9


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OK... no more typing for me.
I hope that helped. If not feel free to ask clarification questions.

Thanks! Like you said, "simple geometry"! (Or was that trig?)

Yes... it is trig.

And it is easy to me, but I use it all the time being a Mechanical Engineering major.

I think computer monitors wide screen type use a 16/10 aspect ratio.