Tom,
I couldn't resist your challenge, especially since it doesn't involve calculus.
Since the height of the screen doesn't change when going between 16:9 and 4:3, we'll need to reduce the 16:9 ratio. For ease of calculating, we'll switch the order of the ratios to height:width, so we now have 3:4 and 9:16.
Reducing the second ratio by a factor of 3 (so the heights match), we get 3:5.333
Now we need to determine the diagonals. The 3:4 aspect ratio is easy, because it creates a classic 3:4:5 triangle. We'll need to actually use the Pythagorean Theorem to find the diagonal of the 3:5.333 aspect ratio:
3^2 + 5.333^2 = 37.444, the square root of which is approximately 6.12.
So we now have 5:6.12 as the ratio between the diagonals of the two aspect ratios. Evaluated, this ratio is 0.817.
To figure out the 4:3 diagonal from the 16:9 diagonal, just multiply the full diagonal by 0.817.
Oh, um... QED, and stuff.
