The radius of the circle doesn’t come into play. This is because instead of enlarging the circle, as you say, what one does is narrow in on smaller and smaller

The radius of the circle doesn’t come into play. This is because instead of enlarging the circle, as you say, what one does is narrow in on smaller and smaller sections of the circle. Given any circle one can narrow in on a small enough piece that, essentially, when looking at it, it will be almost straight. This is true for almost all curves you can think of and draw and is the basis of calculus. Calculus is the study of functions whose graph locally looks like a straight line. Inscribe a 30 sided polygon inside a circle of radius 10 cm. Visually you’ll find it hard to see the difference between the polygon and the circle. Using the formula for the area of a triangle you can calculate the area of the inscribed polygon very easily. This provides an approximation to the area of the circle. Now do this for a 40 sided polygon. Then a 50 sided polygon. A pattern will emerge and one then sees that the limit, which is what happens as the number of sides gets larger and larger without bound, is the familiar formula for the area of a circle. This is how you can prove what the formula for the area of a circle is. You can think of a circle as an infinite sided regular polygon.