Perhaps what you are thinking about is what Archimede's did. Imagine placing "n" points equally spaced around the circumference of the circle, draw lines between neighboring points The total lengths of those lines approximates the circumference of the circle and the larger you take n, the better the approximation is.
Of course, to get any sort of numerical answer, you will need draw lines from those points to the center of the circle so you get n triangles and then use some kind of formulas to relate those lengths to the vertex angle. Archimedes reduced those to square roots and then used a complicated scheme for approximating square roots to any accuracy to show that the circumference is a multiple of the diameter and then get an accurate value for that multiple, basically determining that pi is between 3 and 10/71 and 3 and 1/7. Here is a link to that:
Archimedes' Approximation of Pi
But the basic difficulty is your distinction between "mathematical formulas" and "pure geometry". The ratio of circumference to diameter of a circle is a number, not a geometric object. Whatever method you use, the result will be a "mathematical formula".