Π=3.141592

Why π?

Because it's one of the most useful and beautiful natural constants! It's helpful and important in many fields of Advanced Mathematics, e.g. in statistics or geometry, to get proper results. The number, that is so mysterious because of its irrationality, is also frequently used in science - especially in physics. Scary- and fascinatingly the number π appears in moments you haven't even expected it!

→ Here you get to the related Wikipedia-Article

How to approximate π?

I will put some mathematical strategies to approximate Pi here, when I found something interesting.

First idea:

By graphical intergation of the so called "circle function" ($f(x)={\sqrt {r^{2}-x^{2}}}$), with $r=1$

I. ${\frac {1}{4}}\cdot A_{c}=\int _{0}^{\infty }f(x)\ dx$

Annotation: This integration is quite complex - especially finding a suitable antiderivative - so I will describe a graphical solution! We just have to examine values that fullfill the condition: $0\leqslant x\leqslant 1$, because we want to work with the "positive quarter" of the circle.

II. ${\frac {1}{4}}\cdot A_{c}=\sum _{m=1}^{n}f(m\cdot {\frac {1}{n}})\cdot {\frac {1}{n}}$ with $n\rightarrow \infty$

III. Calculate ${\frac {1}{4}}\cdot A_{c}$with the formular given in II. by setting $n$ to a free chooseable value. (High values recommended!)

Examples

n	0.25·Ac
50	0.774567
150	0.781905
300	0.783675
10,000	0.785347

As can be see in the grid above, ${\frac {1}{4}}\cdot A_{c}$has a realtively constant value of 0.785.

The area of a circle is calculated with the formular $A=\pi r^{2}$, we examined a quarter of a circle with $r=1$

→ $0.785={\frac {1}{4}}\pi$ → $\pi \approx 3.14$