A circle consists of all points at the same distance (the circle’s radius) from its center. In this setting, π is the ratio of the circle’s circumference to its diameter (twice its radius). This definition presupposes the notion of distance and implicitly assumes that we are working in Euclidean geometry. It thus brings us to a practical problem: estimating the circumference of a circular object.
A second definition, also geometric and again involving the circle, concerns the disk, the region inside the circle: π is the ratio of the area of the disk to the square of its radius. It is not obvious that this new π is the same as the previous one (see box), since the first definition concerns length and the second area!
There are many other definitions, but π has been studied and estimated since antiquity because of its connection with the practical problem of calculating circumferences and areas.
Babylon, then Egypt
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