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 |

9 : 8 | √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% |

17.5 : 9 |
√1549 / 35 | 9/8 |
8 |
<0.05% |

18.5 : 9 |
√1693 / 37 | 10/9 |
9 |
<0.09% |

24 : 11 | √697 / 24 | 11/10 | 10 | <0.02% |

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% |

73 : 24 (~3.04 : 1) |
√5905 / 73 | 20/19 | 19 | <0.003% |

25 : 8 | √689 / 25 | 21/20 | 20 | <0.005% |

3.2 : 1 | √281 / 16 | 22/21 | 21 | <0.007% |

29.5 : 9 | √337 / 16 | 23/22 | 22 | <0.005% |

30 : 9 | √109 / 10 | 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% |

51 : 14 (~3.64 : 1) |
√2797 / 51 | 28/27 |
27 | <0.005% |

89 : 24 {~3.71 : 1} |
√8497 / 89 | 29/28 |
28 |
<0.0007% |

34 : 9 |
√337 / 16 | 30/29 |
29 |
<0.07% |

3.84 : 1 | √9841/ 96 | 31/30 | 30 | <0.002% |

43 : 11 (~3.84 : 1) | √1970 / 43 | 32/31 | 31 | <0.005% |

4 : 1 |
√17 / 4 | 33/32 |
32 |
<0.02% |

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 3/2 140/99 |
<0.2% <6% <0.006% |

(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… ADD: 140/99 not easy either)

[*] **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.

…and for completeness d/h = √(1+α²)

A related issue is what the ratio is between the the area and square of the diagonal?

For side lengths **w** and **h** the ratio obviously is wh/d² = wh/(w²+h²) = 1/(w/h + h/w) = 1/(α + 1/α)

Aspect Ratio |
Exact |
Fraction |
Accuracy |

1:1 |
1/2 |
1/2 | exact |

4:3 |
12/25 |
0.48 | exact |

3:2 |
6/13 |
6/13 | exact |

16:9 |
144/337 |
3/7 | <0.3% |

21:9 |
21/58 |
4/11 | <0.5% |

2.39:1 |
2.39/6.7121 |
5/14 | <0.6% |

24:9 |
24/73 |
1/3 | <1.4% |

3:1 |
3/10 |
3/10 | exact |

4:1 |
4/17 |
4/17 | exact |