How is the value of $\pi$ ( Pi ) actually calculated? When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number  and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$.    

But where does this value comes from?

In short How do I derive $\pi$?
There is the famous formula 
$$\dfrac{\pi}{4} = 1 - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{1}{7} \pm \cdots.$$
Unfortunately, this converges rather slowly.  (If you compute to $n$ terms and multiply by $4$, 
you get within roughly $1/n$ of the value of $\pi$; so to get 3 decimal places of accuracy, you'll need to sum something like the first thousand terms of the series.) 

There is the Machin formula
$$\dfrac{\pi}{4} = 4\arctan \dfrac{1}{5} - \arctan \dfrac{1}{239},$$
which can be combined with an infinite series formula for $\arctan$
to provide a much more rapidly convergent infinite series which can be used to compute
$\pi$ to many digits of accuracy.  

The Wikipedia entry provides more details about these and other methods, both historical and contemporary, for computing $\pi$.
$$\pi \;\approx\; 180 \cdot \frac{\sin 10^{-n}}{10^{-n}}$$

where $n$ is a positive real number.
The greater the value of $n$ the more exact 
the value of π becomes.