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The formula for the area of a circle, pi r^2, is traditionally attributed to ancient Greek mathematicians. The method for calculating the area of a circle was likely known in some form by earlier cultures, such as the Egyptians and Babylonians, but it was the Greeks who provided a more rigorous mathematical treatment.
One of the key figures in this development was Archimedes of Syracuse (c. 287-212 BCE). Archimedes used a method involving the approximation of the circle by inscribed and circumscribed polygons to derive the area formula. By increasing the number of sides of these polygons, he could get an increasingly accurate approximation of the circle's area and demonstrated that the area is indeed pi r^2.
While Archimedes did not have the modern notation for π or the concept of limits, his work laid the foundation for later mathematicians to rigorously prove the area formula using calculus and more advanced mathematical techniques.
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