For a right angled triangle, it works the same for 4:3 or 16:9.
For 4:3 its:
actually really easy. 4 squared + 3 squared = hypotenuse squared which is 25. So the square root = 5. So the diagonal is 25% longer than the 4 in 4:3.
There you go. The 16:9 ratio still holds too.
It works like this for 1.77:1
Using the common 1.77:1 screen we use the ratio of 16:9 or Now what you want to know is what the diagonal size is. You need to find the side of the right angled triangle that is unknown. That just happens to be the hypotenuse So lets go back to math class :p
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
16 squared + 9 squared = 337 (or the hypotenuse squared) So to find the diagonal we have to find the square root of 337. The square root of 337 = 18.36
So using inches as a unit, we now you know that a screen with dimensions 16:9 has a diagonal of 18.36". So just figure out how that diagonal compares to the longer side.
So now you know that the diagonal of any 16:9 screen is going to be approximately 1.148 x the length of the longest side, the horizontal side.
Opps. Didn't catch that, I was typing too fast. So just replace the 1.78:1 - 16:9 ratio with 4:3 or 1.33:1.
Really, what I was doing, is showing you that you don't have to do the squares once you have the ratio. Take any 4:3 screen and multiply the width x 1.25 and you'll get the diagonal. Likewise, take a 16:9 screen and multiply the width by 1.147 and you get the diagonal measure.
Or multiply the height of a 4:3 by 1.66 or a 16:9 by 2.04.
Why do I know these ratios? It gives you the ability to figure out your diagonal screen size when given one of your limitations. If you are limited by height before width, like me, I can only make a screen 40 inches high. I have way more than 70 inches across, but with only 40 inches of height, I know that 40 x 2.04 =81.60 inches.
Try it out when fitting a screen for a certain area. Makes things easier to visualize and figure out.