a formula for π 2 (a definition)

There is no easy way for definition of $\pi$ .
One of the most simplest definition of $\pi$ is using the distance.

You must know $\pi$ is equivalent to the length of a semicircular circumference.
That is, $\pi$ is double the length of the circumference of the quadrant.

The graph $C$ is drawn by the function $y=f(x)$ on the interval $x\in[a,b]$ .
$f(x)$ is differentiable and $f'(x)$ is continuous on $[a,b]$ .
Then, the length $l$ of $C$ is
\[ l=\int_a^b \sqrt{1+f'(x)^2}dx .  \]

The function of a circular whose center is origin and radius is 1 is
\[ x^2+y^2=1 .  \]
Thus, the length of quadrant circumference is
\[ l=\int_0^1\frac{1}{\sqrt{1-x^2}}dx  \]
because
\[ y=\sqrt{1-x^2} ,  \]
\[ y'=-\frac{x}{\sqrt{1-x^2}} ,  \]
\[ (y')^2=\frac{x^2}{1-x^2} ,   \]
therefore,
\[ l=\int_0^1\sqrt{1+\frac{x^2}{1-x^2}} dx  . \]

We will define
\[ 2l= 2\int_0^1\frac{dx}{\sqrt{1-x^2}}=\pi  . \]