Pi Formulae

Just some interesting formulae for approximating pi, with graphs. Made for Pi Day 2026.

Formulae

Madhava-Leibniz formula

Dervied from the Taylor series expansion of $arctan(1)$

$\pi=4\sum_{n=1}^\infty \frac{(-1)^n}{2n+1}$

Nilakantha series

Refinement to the Madhava-Leibniz formula.

$\pi=3+4\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n)(2n+1)(2n+2)}$

Ramanujan formula

I don't know how this works. This one is just magic, I think.

$\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty \frac{(4n)!(1103+26390n)}{396^{4n}(n!)^4}$

Building

The project can be built using CMake. All dependencies are included as Git submodules (use a recursive clone to get these).

mkdir build
cd build
cmake ..