A billionth of a century is approximately pi seconds. The diameter of the Earth is roughly half a billion inches.
There are a lot of ways to measure the world and only so many numbers to do it. Every now and then strange coincidences arise. They don’t have any deeper meaning; they don’t indicate some hidden law; they don’t hint at some profound underlying alignment between incongruous realities. They’re just a neat coincidence.
One billionth of a century, affectionately known as a nanocentury, is roughly 3.156 seconds. That number is within one and a half percent of pi (3.1415…), which is interesting but a complete coincidence. Constants like pi appear in a few other mathematical coincidences: pi cubed is extremely close to 31; e cubed is extremely close to 20; pi plus pi plus e is almost exactly 9.
(Incidentally, one millionth of a century is about 52 and a half minutes. This microcentury is apparently the optimal time for a lecture: long enough to get into depth and detail; not so long that people fall asleep.)
Our planet is not a perfect sphere – it’s squished at the poles. If you measure the diameter of the Earth from one pole to the other you get 12713.6 km… or 500,535,433 inches. The inch is supposed to be roughly equivalent to the width of a human thumb, so it is once more complete chance that the two line up so closely.
The Generalist Academy 
Physicists usually say “the number of seconds in a year is pi * 10^7”. In fact it is four times closer to 10^{7.5}, no pi involved, and easier to manipulate algebraically to boot.