# 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 .

• $\pi$ has nothing to do with it. Where did you find that? To find the number of seconds in a year, multiply 1 year by 365.25 days by 24 hours by 60 minutes by 60 seconds. – HDE 226868 Sep 21 '14 at 14:54
• I know, but pi*10^7 is sometimes given as a rough approximation to be used if you dont have a calculator – user58953 Sep 21 '14 at 15:00
• – user58953 Sep 21 '14 at 15:00
• It works fine, but it seems to be a coincidence, unlike, say, the distance the Earth travels in one year. – HDE 226868 Sep 21 '14 at 15:01
• **Why does a nano century equate to $\pi$ seconds? ** Either cesium atoms know about earth's trajectory around sun, can compute $\pi$, and play a trick on us. Or... This is an (approximate) coincidence. Fait vos jeux... – Johannes Sep 21 '14 at 15:44

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.

• obligatory xkcd link – Floris Sep 21 '14 at 14:58
• @Floris: In this case, the xkcd link should really be the answer, not just an obligatory joke comment. – R.. Sep 21 '14 at 19:45
• @R.. Except that the original cartoon didn't have this expression, and Randall subsequently got it wrong by a factor 10 in the subtitle... But yes it's a great one! – Floris Sep 21 '14 at 19:48
• +1, but it would be better if you changed that 365 to 365.25. That value is something every physicist should know. For one thing, 365.25 days is 1/100 of an astronomical century, which is something every astronomer knows by heart. For another, a year of 365.25 years was good enough for humanity for over a thousand years. (And it's obviously still good enough for astronomers!) – David Hammen Sep 21 '14 at 20:15
• @DavidHammen: Most every source I have seen use 365 days & get the same value I have, even Google does it. I will add the extra pair of decimal places to the answer though – Kyle Kanos Sep 21 '14 at 20:25

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.

## protected by Qmechanic♦Sep 21 '14 at 21:34

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