Calculate $\pi$ By Hand?

By hand, it's relatively easy to use the development of the arctangent, and a Machin-like formula:

Machin: $$\frac\pi4=4\arctan\frac15-\arctan\frac1{239}$$

Gauss: $$\frac\pi4=12\arctan\frac1{18}+8\arctan\frac1{57}-5\arctan\frac1{239}$$

I have done it once with Machin's formula and 24 decimals, in a few hours. It's recommended to do it by two methods, to check there is no computation error.

The arctangent is

$$\arctan x=\sum_{k=0}^\infty (-1)^k\frac{x^{2k+1}}{2k+1}$$

Given a number of decimals, you find where to truncate by estimating the rest, and it's easy since it's an alternating series (so the rest is less in absolute value than the first omitted term).

Jean-Claude Arbaut has reminded us of the identity $$ \frac\pi4=4\arctan\frac15-\arctan\frac1{239}. $$ Let us examine that. You learned in high school that $\tan\dfrac\pi4=1$, and that

\begin{align} \tan(\alpha+\beta) & = \dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} \tag 1 \\[10pt] & =\frac{c+d}{1-cd} \end{align}

Thus $$ \arctan c+\arctan d=\alpha+\beta=\arctan\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}=\arctan\frac{c+d}{1-cd} $$

From $(1)$ we get $$ \tan(\alpha+\beta+\gamma+\delta)=\frac{c+d+e+f-cde-cdf-cef-def}{1-cd-ce-cf-de-df-ef+cdef} $$ where $c,d,e,f$ are the respective tangents of $\alpha,\beta,\gamma,\delta$, and hence $$ \tan(4\alpha) = \frac{4\tan\alpha-4\tan^3\alpha}{1 - 6\tan^2\alpha+\tan^4\alpha}. $$ Hence $$ 4\arctan c = \arctan\frac{4c-4c^3}{1-6c^2+c^4}. $$ So $$ 4\arctan\frac15 = \arctan\frac{(4/5)-(4/5^3)}{1-(6/5^2)+ (1/5^4)} = \arctan\frac{480}{476} = \arctan\frac{120}{119}. $$ Next we look at \begin{align} & 4\arctan\frac15 - \arctan\frac{1}{239} = \arctan\frac{120}{119} - \arctan\frac{1}{239} \\[15pt] = {} & \arctan\frac{(120/119)-(1/239)}{1+(120/119)(1/239)} \\[15pt] = {} & \arctan\frac{28561}{28561} = \arctan 1 = \frac\pi4. \end{align}

The fastest known formula for calculating the digits of pi is Chudnovsky formula: $$\frac{1}{\pi}=12 \sum_{k=0}^\infty \frac{(-1)^k (6k)! (163 \cdot 3344418k + 13591409)}{(3k)! (k!)^3 640320^{3k+1.5}}$$ This formula is used to create world record for the most digits of pi. This formula rapidly converges and it needs 3-4 terms to yield good approximation of pi which is possible by hand.