Obviously it is easy to calculate the diagonal of e.g. a sensor from the width and height with Pythagoras.
And a bit more tedious the other way round using the aspect ratio. [*]
But for common aspect ratios it is even simpler, 1+1/n fractions within less than 0.4%
For various aspect ratios (common in bold), here the ratio of diagonal versus long edge [*], approximated by a fraction:
| Aspect Ratio | Exact | Fraction | 1+1/n | Accuracy |
| 8 : 9 | √145 / 9 | 4/3 | 3 | <0.3% |
| 4 : 3 | 5/4 | 5/4 | 4 | Exact |
| 3 : 2 | √13 / 3 | 6/5 | 5 | <0.2% |
| 15 : 9 | √34 / 5 | 7/6 | 6 | <0.05% |
| 16 : 9 | √337 / 16 | 8/7 | 7 | <0.4% |
| 21 : 9 | √58 / 7 | 12/11 | 11 | <0.3% |
| 21.15 : 9 or 2.35 : 1 | √2609 / 47 | 13/12 | 12 | <0.4% |
| 21.51 : 9 or 2.39 : 1 | √67121 / 239 | 13/12 | 12 | <0.06% |
| 21.6 : 9 or 2.40 : 1 | 13/12 | 13/12 | 12 | Exact |
| 2.5 : 1 | √29 / 5 | 14/13 | 13 | <0.01% |
| 22.95 : 9 or 2.55 : 1 | √3001 / 51 | 15/14 | 14 | <0.3% |
| 23.94 : 9 or 2.66 : 1 | √20189 / 133 | 15/14 | 14 | <0.3% |
| 24 : 9 | √73 / 8 | 15/14 | 14 | <0.4% |
| 2.7 : 1 | √829 / 27 | 16/15 | 15 | <0.03% |
| 25 : 9 | √706 / 25 | 17/16 | 16 | <0.03% |
| 26 : 9 | √757 / 26 | 18/17 | 17 | <0.06% |
| 3 : 1 | √10 / 3 | 19/18 | 18 | <0.2% |
| 3.2 : 1 | √281 / 16 | 22/21 | 21 | <0.007% |
| 30 : 9 | √1.09 | 24/23 | 23 | <0.06% |
| 24 : 7 (~3.43 :1 ) | 25/24 | 25/24 | 24 | Exact |
| 3.5 : 1 | √53 / 7 | 26/25 | 25 | <0.002% |
| 32 : 9 | √1105 / 26 | 27/26 | 26 | <0.04% |
Note that these are not only fractions but all of the simple form 1 + 1/n, so hardly gets more easy….
So I wonder why this is not ‘common knowledge’. Obviously it (for most) are just numerical coincidences, but an odd fun-fact non the less.
For completeness of the common aspect ratios:
| Aspect Ratio | Exact | Fraction | Accuracy |
| 1:1 | √2 | 24/17 3/2 |
<0.2% <6% |
(But 24/17 is just as hard to remember as a few decimals of √2, and 6% is a bit much, so not really useful…)
[*] d/w= √(1+1/α²) Follows from d²=w²+h² with aspect ratio α=w/h where w and h are the long and short edge lengths.