# 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 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