Diagonal to width, simple fractions. Different aspect ratios

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.