After fiddling around a bit, eventually found another way of getting an approximation for the royal cubit. I say approximation because, compared to the other methods, it’s only accurate to about 3 decimal places.

The formula looks like this:

or

What I like about it is how it uses what are arguably the most famous numbers of all time in all of mathematics (yes, √2, you’ve just been demoted) and wraps them all nicely to produce a reasonable figure for the royal cubit.

Not quite as elegant as Euler’s famous formula, but still interesting.

By way of comparison, here’s the others:

Method | Value | Difference from π/6 | Abs difference |
---|---|---|---|

π/6 | 0.52359877559830 | 0.00000000000000 | 0.00000000000000 |

φ²/5 | 0.52360679774998 | 0.00000802215168 | 0.00000802215168 |

Ave Year | 0.52360779708785 | 0.00000902148956 | 0.00000902148956 |

π – φ² | 0.52355866483991 | -0.00004011075839 | 0.00004011075839 |

10²/π²φ²ⅇ² | 0.52376444099003 | 0.00016566539172 | 0.00016566539172 |

If we really want to keep root 2 happy, we could rewrite it like this, and cheat a bit.

Where tau τ is equal to 2 π.