Just as 4:3s were starting to be phased out, I realized that this stuff is just so easy to figure for a 4:3 because the two sides and the diagonal make a 3:4:5 triangle. The easiest right triangle to deal with. Let's look at triangles for a moment.

For instance, if a set has a 35" diagonal (that is 7x5), its image height will be 21" (3x7 and its width 28" (4x7). Diagonal x 0.8 = width; Diagonal x 0.6 = height. Not so difficult numbers. That's a 4:3.

The formula is that famous one parodied by the joke with a punch line something like "the squaw of the hippopotamus is equal to the sum of the squaws of the other two hides." That is, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is the Pythagorean Theorem showing the relationships among the three sides of a triangle with a 90 degree corner.

For a 3:4:5 triangle, 3 squared = 9, four squared = 16, 9 + 16 = 25. It just so happens that 5 squared = 25. I don't know of any other numbers for which there is so simple a relationship, but 16 and 9 are NOT two of them!

After doing the math...
If the height is 9 and the width is 16, the diagonal is 18 3/8 ... more or less (square root of sum of 81 + 256 = 337; sq rt of 337 = 18.358).

More math says that if you have the diagonal (D), the height will be D x 9 / 18.358 or 0.49D. The width will be D x 16 / 18.358 or 0.872D.

I find it easier to communicate this size thing by considering that if you change from a 4:3 to a 16:9, people will look like they are the same size if their height remains the same. TVs of different aspect ratios can appear to have the same size people, but the 16:9 will show a bit more out to the sides. So...what size 16:9 is going to give you the same size person as a 4:3?

A 32" 4:3 will have a height of 32" x 0.6 = 19.2". A 34" 16:9 will have a height of (give me a couple of minutes here) 34 x .49 = 16.66".

So how big would a 16:9 have to be so people look the same size as a 4:3? A 16:9 with a height of 19.2" would have a diagonal of 19.2 x 18.358 / 9 = 39.163". The inconvenient number to remember here is that a 16:9 will have the same size (height) picture (people) if it is 1.15 times the diagonal of a 4:3.

Lemme know if you can figure out an easy way to remember these numbers. I just had to figure them out again to answer you. Maybe I'll put 'em in my Palm Pilot.