Rational approximations are always useful when approximating real numbers.

​

While 22/7 is an okay sort-of-but-not-so-good approximation of π, 355/113 is significantly better. The relative error of the first is 0.040%, while the relative error of the second is 0.0000085%, comparing the approximation at nine significant digits with 3.14159292 and π rounded to nine digits 3.14159265. Ton contrast, 22/7 = 3.14285714.

​

The best approximation for the length of a year in terms of days is to:

1. have each year be 365 days,

2. unless the year is a multiple of four, in which case it is a leap year with 366 days,

3. unless it is also a multiple of 128, in which case it is a 365-day year.

This gives the average number of days in each year to be 46751/128 = 365.2421875 days. A tropical year is 365.24219 days, so the error is less than one-quarter of one second per year.

​

A lunar month is approximation 29.530588 days, and a reasonable approximation is 29 + 26/49, so if 26 out of every 49 months is 30 days, and the remaining 23 are 29 days, this will be a good enough approximation that this will lose only slightly more than two seconds per lunar month. After a century, the synchronization would be off by less than three-quarters of an hour.

This could be implemented by having one 15-month period alternating 30-29-30-...-30-29-30,

followed by two 17-month periods alternating in the same pattern,

totaling 49 months, with 8 + 9 + 9 = 26 30-day months.

Of course, this is only an approximation because there is significant variation throughout the year as to how long a lunar month is, so it may happen that there are four months in a row that have 30 days, and it is also possible to have a lunar month being only 28 days. Over the long term, however, this should be a reasonable approximation.

​

A really good approximation to an equilateral triangle is to have the base 8 and the height 7.

The hypotenuse is the square root of 65, or 8.0623, so a relative error of less than 1% and the angles are 60.255°, 60.255° and 59.49°.

If you want greater accuracy, use a base of 22 and a height of 19

The hypotenuse is the square root of  482, or 21.9545, so a relative error of 0.2% and the angles are 59.93°, 59.93° and 60.14°.

If you require even greater accuracy, use a base of 52 and a height of 45.

The hypotenuse is the square root of 2701, or 91.97, so a relative error of 0.0555% and the angles are 59.98°, 59.98° and 60.04°

​

The first approximation is an excellent means of turning square grid paper into a hexagonal grid. Depending on the size of the grid, either of these solutions is quite acceptable as an approximation to a hexagonal grid: