Calculate $\pi$ By Hand? All over the internet the only hand equation i found was 
$$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$
But this takes something like a thousand iterations to get to four digits, is there a better way to calculate pi by hand?

One easy-to-understand improvement to your method, which I I don't see used much, is:
$$\pi/6 = \arctan \left ( \frac{\sqrt{3}}{3} \right ) \\
 = \int_0^{\frac{\sqrt{3}}{3}} \frac{1}{1+x^2} dx \\
= \sum_{n=0}^\infty \frac{(-1)^{n} \left ( \frac{\sqrt{3}}{3} \right )^{2n+1}}{2n+1} \\
= 3^{-1/2} \sum_{n=0}^\infty \frac{(-1)^{n} 3^{-n}}{2n+1}.$$
Consequently we have
$$\pi = 2 \sqrt{3} \sum_{n=0}^\infty \frac{(-1)^{n} 3^{-n}}{2n+1} \\
\approx 2 \sqrt{3} \sum_{n=0}^N \frac{(-1)^{n} 3^{-n}}{2n+1} \\
= 2 \sqrt{3} \left ( 1 - \frac{1}{9} + \frac{1}{45} - \frac{1}{189} + \dots \right )$$
This gives each decimal digit in slightly fewer than $\log_3(10) \approx 2.1$ steps, provided you can accurately estimate $\sqrt{3}$ to do the final multiplication.
