$1.$ Viète's formula for $\pi$ is easy to derive without trigonometry or calculus. Starting with a square inscribed in a circle of unit diameter, the number of sides of the inscribed polygon is repeatedly doubled. https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula
$2.$ Viète's formula (before taking the limit as $k$ goes to infinity) gives the perimeter of a $2^{k+1}$-sided regular polygon inscribed in a circle of unit diameter.
$3.$ Because the square of a rational number is rational, the square root of an irrational number is irrational. So, working from right to left in the nested square roots of Viète's formula, the perimeters of ALL the polygons are irrational.
$4.$ As $k$ becomes very large, the perimeter of the polygon converges to $\pi$. Therefore $\pi$ is irrational.