Approximation to the number of seconds in a year? Is there any mathematical formula which shows that there are approximately $\pi \times 10^7$ seconds in one year. I understand that the pi is probably due to the earth's circular orbit, but am not sure where the rest could come from .
 A: The observation that "π seconds is a nanocentury" is attributed to Tom Duff, who is known to computer programmers as the inventor of "Duff's Device".  There's nothing magical about the fact that 1/10,000,000 of a planetary orbit should equal roughly π/86400 of a planetary day (not the same thing as a planetary rotation, btw, since the orientation of the side of the earth facing the sun changes during the orbit); if the earth turned at a slightly different speeds, then one might say "e seconds is a nanocentury", but it would be no more meaningful.
A: It's a unit conversion:
$$
1\,{\rm yr}=\frac{365\,{\rm days}}{\rm year}\times\frac{24\,{\rm hrs}}{1\,{\rm day}}\times\frac{3600\,{\rm sec}}{1\,{\rm hour}}=3.1556926\times10^7\,{\rm sec}
$$
Since $3.1557$ is (somewhat) close to $\pi\sim3.1416$, we use the approximation you cite.
Technically, the year is actually 365.25 days long, so using that gives a slightly better approximation that gets one to $3.15576\times10^7\,{\rm sec}$, though most sources I've seen simply use 365 days. Both values are still less than half a percent off of the $\pi\cdot10^7$ value.
