Once has been defined as the ratio circumference to diameter, the area of a circle must be .
A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas.
A circle is cut up into 6 sectors which are then rearranged into a near rectangle,
What doesn’t look too close to a rectangle at 6 sectors looks better at 26:
As the number of sectors becomes very big the shape becomes indistinguishable from a rectangle and the argument is complete.
I think that Archimedes used such arguments, or, proof by exhaustion, in some of his solid geometry work.