34 Ways to calculate the Royal Cubit

Updated versions of the different ways of approximating the cubit. Also includes separate table for Grand Metre (1 plus royal cubit) == 1.5236…. == 1.524.

These are done with pi, phi, e, roots and powers (usually of basic primes), as well as ln, log, sin, cos and tan.

See square roots, cube roots and ln(4) for formulas not shown below.

Changelog at the bottom.See also The Magical Mystical Royal Cubit for the Pretty Picture version.

MethodValueDifference from π/6Abs differenceRound
π/60.5235987760.0000000000.0000000000.5236
Very close
((6√2/10)² + (6/100)²+(8√2/10000)²)²0.5235988120.0000000370.0000000370.5236
cube roots0.5236003500.0000015750.0000015750.5236
((6√2/10)² + (6/100)²)²0.523596960-0.0000018160.0000018160.5236
(7π/5e)²/50.523596637-0.0000021380.0000021380.5236
28φπ/100e0.5236017170.0000029420.0000029420.5236
(φe)/8.40.5236038560.0000050800.0000050800.5236
ln(4)0.523591499-0.0000072770.0000072770.5236
((1 + π)/e) - 10.5236067910.0000080150.0000080150.5236
φ²/50.5236067980.0000080220.0000080220.5236
Close
π – (7π/5e)²0.5236094680.0000106920.0000106920.5236
(eφ³)/7π0.5236118780.0000131030.0000131030.5236
(10φ)/(11e + 1)0.5236169530.0000181770.0000181770.5236
tan(2φπe)0.523569002-0.0000297740.0000297740.5236
π – φ²0.523558665-0.0000401110.0000401110.5236
√(√5/(3e))0.5236421930.0000434170.0000434170.5236
e(2√2 - φ)/2π0.5236499490.0000511740.0000511740.5236
Less close
10e/(36³√3)0.523542042
-0.000056730.000056733
0.5235
ln(10)/φe0.523520348-0.000078430.0000784270.5235
e/(3√3)0.523133582-0.0004651940.0004651940.523
square roots0.523403737-0.0001950390.0001950390.523
1/(√(ln365.25/φ))0.5236563730.0000575970.0000575970.524
2√5/πe0.5236856130.0000868380.0000868380.524
1/log(φ²π³)0.5237179010.0001191250.0001191250.524
(10/φπⅇ)²0.5237644410.0001656650.0001656650.524
(10√2)/270.5237828010.0001840250.0001840250.524
cos(π(4φe+1))0.5238087940.0002100190.0002100190.524
(∛3 ∛5 ∛7)/90.5241882200.0005894440.0005894440.524
(∛5φ²)/πⅇ0.5242288610.0006300860.0006300860.524
sin((φ²πe√2)0.5242537150.0006549400.0006549400.524
(φ)/(e + 1/e)0.5242869210.0006881450.0006881450.524
(√2+√3)/60.5243773950.0007786190.0007786190.524
√(7√2)/60.5243910470.0007922720.0007922720.524
π/(φ³√2)0.5244111940.0008124190.000812419.0524

Grand metre:

Grand Metre comparisons, 2018-12-04

MethodValueDifference from 1 + π/6Abs difference
1 + π/61.523598775598300.000000000000000.00000000000000
approx 1.5236
(1 + π)/e1.523606790962400.000008015364070.00000801536407
π - φ1.52355866483990-0.000040110758400.00004011075840
approx 1.524
(e√2/π) + 3φe/14π1.523659651828300.000060876230040.00006087623004
π^(1/e)1.523671054858900.000072279260630.00007227926063
1.2345²1.523990250000000.000391474401700.00039147440170
(1.1111²)²1.524096937357100.000498161758810.00049816175881
∛3∛5/φ1.524202885401600.000604109803320.00060410980332

Putting things in perspective

If we for example look at the φe/8.4 formula, that differs from π/6 by 0.00000508028836 metres. That sort of number evokes three reactions… the mathematicians will say it’s not equal to ₢, engineers will say its “close enough for practical purposes” and the rest will have some variation of WTF or “what the heck does that mean?”

So to put things in perspective, I’ve converted the differences to microns and used some nice comparisons provided to WikiPedia to help us visualise just how small those differences are. Remember the royal cubit was typically used for building projects or land surveying, where such differences are not going to make a difference.

π/60.5235987760.000000000
Very close
((6√2/10)² + (6/100)²+(8√2/10000)²)²0.5235988120.0000000370.04length of a lysosome
cube roots0.5236003500.0000015751.57anthrax spore
((6√2/10)² + (6/100)²)²0.5235969600.0000018161.82anthrax spore
(7π/5e)²/50.5235966370.0000021382.14length of an average E. coli bacteria
28φπ/100e0.5236017170.0000029422.94size of a typical yeast cell
(φe)/8.40.5236038560.0000050805.08length of a typical human spermatozoon's head
ln(4) formula0.5235914990.0000072777.28diameter of human red blood cells
((1 + π)/e) - 10.5236067910.0000080158.02size of a ground-level fog or mist droplet
φ²/50.5236067980.0000080228.02size of a ground-level fog or mist droplet
Close
π – (7π/5e)²0.5236094680.00001069210.69width of cotton fibre
(eφ³)/7π0.5236118780.00001310313.10width of silk fibre
(10φ)/(11e + 1)0.5236169530.00001817718.18about minimum width of a strand of human hair
tan(2φπe)0.5235690020.00002977429.77length of a human skin cell
π – φ²0.5235586650.00004011140.11less than typical length of a human liver cell, an average-sized body cell
√(√5/(3e))0.5236421930.00004341743.42less than typical length of a human liver cell, an average-sized body cell
e(2√2 - φ)/2π0.5236499490.00005117451.17typical length of a human liver cell, an average-sized body cell
Less close
10e/(36³√3)0.5235420420.00005673356.73less than length of a sperm cell
ln(10)/φe0.5235203480.00007842778.43thickness of sheet of paper
e/(3√3)0.5231335820.000465194465.19less than average length of a grain of sand
square roots0.5234037370.000195039195.04typical length of Paramecium caudatum, a ciliate protist
1/(√(ln365.25/φ))0.5236563730.00005759757.60less than length of a sperm cell
2√5/πe0.5236856130.00008683886.84less than smallest distance that can be seen with the naked eye
1/log(φ²π³)0.5237179010.000119125119.13diameter of a human ovum
(10/φπⅇ)²0.5237644410.000165665165.67length of the largest sperm cell in nature, belonging to the Drosophila bifurca fruit fly
(10√2)/270.5237828010.000184025184.03less than typical length of Paramecium caudatum, a ciliate protist
cos(π(4φe+1))0.5238087940.000210019210.02typical length of Paramecium caudatum, a ciliate protist
(∛3 ∛5 ∛7)/90.5241882200.000589444589.44over 0.5 mm
(∛5φ²)/πⅇ0.5242288610.000630086630.09over 0.6 mm
sin((φ²πe√2)0.5242537150.000654940654.94over 0.6 mm
(φ)/(e + 1/e)0.5242869210.000688145688.15over 0.6 mm
(√2+√3)/60.5243773950.000778619778.62over 0.7 mm
√(7√2)/60.5243910470.000792272792.27over 0.7 mm
π/(φ³√2)0.5244111950.000812419812.42over 0.8 mm

2018-11-26: added (e φ³)/(7π), as well as ((1+π)/e)-1, from Grand Metre table to cubit table, since it is so accurate.
2018-11-28: added (φe)/8.4, (10φ)/(11e + 1), and (φ)/(e + 1/e). Added 1/(√(ln365.25/φ)).
2018-12-02: Added 28φπ/100e. I’ve actually done the φπ/e sum many times, but tonight the lightbulb went on and I thought, hey, that 1.87 answer looks a lot like the value of 0.01875 for the length of a digit … I wonder what happens if I divide by 100 (to get decimal point in right place, else just skip that and work in cm) and multiply by 28, the number of digits in a royal cubit …. and voila … very nice answer. I suppose we could dub this the “royal digit” to differentiate it from the normal digit. The construction of the cubit is very complex, the sample we have in Turin has digits of different length on the last palm.
I also found another approximation using root(2π²)/2φ³ but this would be the worst of the lot so not adding it at the moment … comes out around 0.5244112.
2018-12-03: Added formulas based on 5φe/7π, split table into categories.
2018-12-04: Added pi^(1/e) to Grand Metre section, rearranged that table a bit.
2018-12-08: Added 2√5 / φe and 10√2 / 3³
2018-12-10: Added value from solving for x in  ln(4) formula: ln(4) + x = 1/x