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.

The construction uses the radius of the circle as a parameter. If the radius is bigger, then each piece is bigger.

Imagine that you have this construction drawn as a svg file in the computer, a really big screen, and you can use the zoom. The size of the pieces will match the size of the circle.

For this sort of math (theoretical), "practically straight" is never straight enough.

And your approach is close enough to calculus that "totally naive..." seems a tad too modest.

I think you could get close by making a lot of pie slices, lining them up in a row, and then flipping every other one upside-down. Then you'd be able to put them back together into a rectangle.

The problem with our plan, I guess, is that you'd always be close at every pie slice size, never exactly there. These guys have figured out how to do it exactly, with a finite number of slices, instead of approaching it in the limit of infinity slices.

Math isn't concerned with "practically straight", so it won't make a difference.

For example, consider the number 2 and 2.000000000001. To engineers and scientists, these two numbers are basically the same, and there's no practical difference. To a mathematician, the former is an integer, and the latter isn't - you could add as many zeros to the second number as you want, and the difference won't go away, unless you add an infinite number of zeros, at which point it becomes identical to 2.

I think EVERY circle fulfills this condition ;)