Consider two series of regular polygons inscribed and circumscribed to the circle.

If a_1, a_2, a_3...a_n are the areas of n edges regular polygons inscribed in the circle and  A₁, A₂, A₃...A_n   those of circumscribed regular polygons, it's possible to demonstrate two series be contiguous series of real numbers.

So we can give the following definition:

Circle area is the number fixed by two contiguous series of real numbers corresponding to areas of regular polygons inscribed and circumscribed to the circle.