{"id":4586,"date":"2022-08-01T12:10:38","date_gmt":"2022-08-01T10:10:38","guid":{"rendered":"https:\/\/eskerahn.dk\/?p=4586"},"modified":"2022-10-17T07:05:29","modified_gmt":"2022-10-17T05:05:29","slug":"fun-fact-diagonal-to-width-simple-fractions-different-aspect-ratios","status":"publish","type":"post","link":"https:\/\/eskerahn.dk\/?p=4586","title":{"rendered":"Diagonal to width, simple fractions. Different aspect ratios"},"content":{"rendered":"

Obviously it is easy to calculate the diagonal of e.g. a sensor from the width and height with Pythagoras.
\nAnd a bit more tedious the other way round using the aspect ratio. [*]
\nBut for common aspect ratios it is even simpler, 1+1\/n fractions within less than 0.4%<\/p>\n

\"\"<\/a><\/p>\n

<\/p>\n

For various aspect ratios (common in bold), here the ratio of diagonal versus long edge [*], approximated by a fraction:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Aspect Ratio<\/strong><\/td>\nExact<\/strong><\/td>\nFraction<\/strong><\/td>\n1+1\/n<\/strong><\/td>\nAccuracy<\/strong><\/td>\n<\/tr>\n
9 : 8<\/span><\/td>\n\u221a145 \/ 9<\/td>\n4\/3<\/td>\n3<\/td>\n<0.3%<\/td>\n<\/tr>\n
4 : 3<\/strong><\/td>\n5\/4<\/td>\n5\/4<\/strong><\/td>\n4<\/strong><\/td>\nExact<\/span><\/strong><\/td>\n<\/tr>\n
3 : 2<\/strong><\/td>\n\u221a13 \/ 3<\/td>\n6\/5<\/strong><\/td>\n5<\/strong><\/td>\n<0.2%<\/td>\n<\/tr>\n
15 : 9<\/td>\n\u221a34 \/ 5<\/td>\n7\/6<\/td>\n6<\/td>\n<0.05%<\/td>\n<\/tr>\n
16 : 9<\/strong><\/td>\n\u221a337 \/ 16<\/td>\n8\/7<\/strong><\/td>\n7<\/strong><\/td>\n<0.4%<\/td>\n<\/tr>\n
17.5 : 9
\n<\/strong><\/td>\n		\u221a1549 \/ 35<\/td>\n9\/8
\n<\/strong><\/td>\n		8
\n<\/strong><\/td>\n		<0.05%<\/td>\n<\/tr>\n
18.5 : 9
\n<\/strong><\/td>\n		\u221a1693 \/ 37<\/td>\n10\/9
\n<\/strong><\/td>\n		9
\n<\/strong><\/td>\n		<0.09%<\/td>\n<\/tr>\n
24 : 11<\/span><\/td>\n\u221a697 \/ 24<\/td>\n11\/10<\/td>\n10<\/td>\n<0.02%<\/td>\n<\/tr>\n
21 : 9<\/strong><\/td>\n\u221a58 \/ 7<\/td>\n12\/11<\/strong><\/td>\n11<\/strong><\/td>\n<0.3%<\/td>\n<\/tr>\n
21.15 : 9 or 2.35 : 1<\/strong><\/td>\n\u221a2609 \/ 47<\/td>\n13\/12<\/strong><\/td>\n12<\/strong><\/td>\n<0.4%<\/td>\n<\/tr>\n
21.51 : 9 or 2.39 : 1<\/strong><\/td>\n\u221a67121 \/ 239<\/td>\n13\/12<\/strong><\/td>\n12<\/strong><\/td>\n<0.06%<\/td>\n<\/tr>\n
21.6 : 9 or 2.40 : 1<\/strong><\/td>\n13\/12<\/td>\n13\/12<\/strong><\/td>\n12<\/strong><\/td>\nExact<\/span><\/strong><\/td>\n<\/tr>\n
2.5 : 1<\/td>\n\u221a29 \/ 5<\/td>\n14\/13<\/td>\n13<\/td>\n<0.01%<\/td>\n<\/tr>\n
22.95 : 9 or 2.55 : 1<\/strong><\/td>\n\u221a3001 \/ 51<\/td>\n15\/14<\/strong><\/td>\n14<\/strong><\/td>\n<0.3%<\/td>\n<\/tr>\n
23.94 : 9 or 2.66 : 1<\/strong><\/td>\n\u221a20189 \/ 133<\/td>\n15\/14<\/strong><\/td>\n14<\/strong><\/td>\n<0.3%<\/td>\n<\/tr>\n
24 : 9<\/strong><\/td>\n\u221a73 \/ 8<\/td>\n15\/14<\/strong><\/td>\n14<\/strong><\/td>\n<0.4%<\/td>\n<\/tr>\n
2.7 : 1<\/span><\/td>\n\u221a829 \/ 27<\/td>\n16\/15<\/td>\n15<\/td>\n<0.03%<\/td>\n<\/tr>\n
25 : 9<\/td>\n\u221a706 \/ 25<\/td>\n17\/16<\/td>\n16<\/td>\n<0.03%<\/td>\n<\/tr>\n
26 : 9<\/td>\n\u221a757 \/ 26<\/td>\n18\/17<\/td>\n17<\/td>\n<0.06%<\/td>\n<\/tr>\n
3 : 1<\/strong><\/td>\n\u221a10 \/ 3<\/td>\n19\/18<\/strong><\/td>\n18<\/strong><\/td>\n<0.2%<\/td>\n<\/tr>\n
73 : 24 (~3.04 : 1)
\n<\/span><\/td>\n		\u221a5905 \/ 73<\/td>\n20\/19<\/td>\n19<\/td>\n<0.003%<\/td>\n<\/tr>\n
25 : 8<\/span><\/td>\n\u221a689 \/ 25<\/td>\n21\/20<\/td>\n20<\/td>\n<0.005%<\/td>\n<\/tr>\n
3.2 : 1<\/td>\n\u221a281 \/ 16<\/td>\n22\/21<\/td>\n21<\/td>\n<0.007%<\/td>\n<\/tr>\n
29.5 : 9<\/span><\/td>\n\u221a337 \/ 16<\/td>\n23\/22<\/td>\n22<\/td>\n<0.005%<\/td>\n<\/tr>\n
30 : 9<\/td>\n\u221a109 \/ 10<\/td>\n24\/23<\/td>\n23<\/td>\n<0.06%
\n<\/strong><\/span><\/td>\n<\/tr>\n
24 : 7 (~3.43 :1 )<\/td>\n25\/24<\/td>\n25\/24<\/td>\n24<\/td>\nExact<\/strong><\/span><\/td>\n<\/tr>\n
3.5 : 1<\/td>\n\u221a53 \/ 7<\/td>\n26\/25<\/td>\n25<\/td>\n<0.002%
\n<\/strong><\/span><\/td>\n<\/tr>\n
32 : 9<\/strong><\/td>\n\u221a1105 \/ 26<\/td>\n27\/26<\/strong><\/td>\n26<\/strong><\/td>\n<0.04%<\/td>\n<\/tr>\n
51 : 14 (~3.64 : 1)<\/span>
\n<\/strong><\/td>\n		\u221a2797 \/ 51<\/td>\n28\/27
\n<\/strong><\/td>\n		27<\/td>\n<0.005%<\/td>\n<\/tr>\n
89 : 24 {~3.71 : 1}<\/span>
\n<\/strong><\/td>\n		\u221a8497 \/ 89<\/td>\n29\/28
\n<\/strong><\/td>\n		28
\n<\/strong><\/td>\n		<0.0007%<\/td>\n<\/tr>\n
34 : 9
\n<\/strong><\/td>\n		\u221a337 \/ 16<\/td>\n30\/29
\n<\/strong><\/td>\n		29
\n<\/strong><\/td>\n		<0.07%<\/td>\n<\/tr>\n
3.84 : 1<\/span><\/td>\n\u221a9841\/ 96<\/td>\n31\/30<\/td>\n30<\/td>\n<0.002%<\/td>\n<\/tr>\n
43 : 11 (~3.84 : 1)<\/span><\/td>\n\u221a1970 \/ 43<\/td>\n32\/31<\/td>\n31<\/td>\n<0.005%<\/td>\n<\/tr>\n
4 : 1<\/strong><\/td>\n\u221a17 \/ 4<\/td>\n33\/32<\/strong><\/td>\n32<\/strong><\/td>\n<0.02%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Note that these are not only fractions but all of the simple form 1 + 1\/n, so hardly gets more easy….
\nSo I wonder why this is not ‘common knowledge’. Obviously it (for most) are just numerical coincidences, but an odd fun-fact non the less.<\/p>\n

 <\/p>\n

For completeness of the common aspect ratios:<\/p>\n\n\n\n\n
Aspect Ratio<\/strong><\/td>\n\u00a0Exact<\/strong><\/td>\nFraction<\/strong><\/td>\nAccuracy<\/strong><\/td>\n<\/tr>\n
1:1<\/strong><\/td>\n\u221a2<\/strong><\/td>\n24\/17
\n3\/2
\n140\/99<\/td>\n		<0.2%
\n<6%<\/span>
\n<0.006%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

(But 24\/17 is just as hard to remember as a few decimals of \u221a2<\/strong>, and 6% is a bit much, so not really useful… ADD: 140\/99 not easy either)<\/p>\n

 <\/p>\n

[*] d\/w= \u221a(1+1\/\u03b1\u00b2)<\/strong> Follows from d\u00b2=w\u00b2+h\u00b2 with aspect ratio \u03b1=w\/h where w and h are the long and short edge lengths.
\n…and for completeness\u00a0 d\/h = \u221a(1+\u03b1\u00b2)<\/p>\n

\n

A related issue is what the ratio is between the the area and square of the diagonal?<\/p>\n

For side lengths w<\/strong> and h<\/strong> the ratio obviously is wh\/d\u00b2 = wh\/(w\u00b2+h\u00b2) = 1\/(w\/h + h\/w) = 1\/(\u03b1 + 1\/\u03b1)<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
Aspect Ratio<\/strong><\/td>\n\u00a0Exact<\/strong><\/td>\nFraction<\/strong><\/td>\nAccuracy<\/strong><\/td>\n<\/tr>\n
1:1<\/strong><\/td>\n1\/2<\/strong><\/td>\n1\/2<\/td>\nexact<\/td>\n<\/tr>\n
4:3<\/strong><\/td>\n12\/25<\/strong><\/td>\n0.48<\/td>\nexact<\/td>\n<\/tr>\n
3:2<\/strong><\/td>\n6\/13<\/strong><\/td>\n6\/13<\/td>\nexact<\/td>\n<\/tr>\n
16:9<\/strong><\/td>\n144\/337<\/strong><\/td>\n3\/7<\/td>\n<0.3%<\/td>\n<\/tr>\n
21:9<\/strong><\/td>\n21\/58<\/strong><\/td>\n4\/11<\/td>\n<0.5%<\/td>\n<\/tr>\n
2.39:1<\/strong><\/td>\n2.39\/6.7121<\/strong><\/td>\n5\/14<\/td>\n<0.6%<\/td>\n<\/tr>\n
24:9<\/strong><\/td>\n24\/73<\/strong><\/td>\n1\/3<\/td>\n<1.4%<\/td>\n<\/tr>\n
3:1<\/strong><\/td>\n3\/10<\/strong><\/td>\n3\/10<\/td>\nexact<\/td>\n<\/tr>\n
4:1<\/strong><\/td>\n4\/17<\/strong><\/td>\n4\/17<\/td>\nexact<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

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%<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7,13,6,4],"tags":[],"class_list":["post-4586","post","type-post","status-publish","format-standard","hentry","category-compact-cameras","category-math","category-phonesphablets","category-various-tech-stuff"],"_links":{"self":[{"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/posts\/4586","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4586"}],"version-history":[{"count":9,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/posts\/4586\/revisions"}],"predecessor-version":[{"id":4725,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=\/wp\/v2\/posts\/4586\/revisions\/4725"}],"wp:attachment":[{"href":"https:\/\/eskerahn.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4586"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4586"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/eskerahn.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4586"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}