Yesterday March 14 was dubbed Pi Day. It also happens that I started this website and blog one year ago ,and it’s Albert Einstein ‘s day of birth.

Here are some facts ,relations and formulas related to the number Pi.

Approximate value of Pi to 1500 digits (done with Mathematica);

The Euler identity or formula ,which mathematicians like very much and find beautiful ,containing five math constants:

Here is a quick derivation of a known formula involving Pi:

where ln is the natural logarithm with base e = 2.7182818284590452354.

Two integrals involving Pi (done and verified with Mathematica):

And here is a brief study I made of a series of formulas involving Pi and the inverse trigonometric function arctangent (done with the Texas Instruments TI 92 Plus calculator and Mathematica) :

The general form for these relations is:

Using the Mathematica function FindRoot , we find two values for f(x) (with x >-1 and x< -1):

Thus we may deduce the following values :

These pi formulas resemble or seem to be related to the Machin-Like Formulas or to the more general formulas :

which have been studied in the past.

I have also considered the study of other formulas involving Pi and other inverse trigonometric functions . Maybe I will publish or write more about these subjects in the future.