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How did Archimedes use the method of exhaustion to calculate areas?

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Infinity — infinity Mathematical symbol representing the concept of infinity.

Archimedes used the method of exhaustion to calculate the areas and volumes of curved figures by employing a rigorous, logical procedure that avoided the use of actual infinity or infinitesimals.12

Instead of treating a figure as being composed of an infinite number of parts, Archimedes approximated the area of a shape by inscribing and circumscribing it with polygons. By successively increasing the number of sides of these polygons, he could "exhaust" the area between the polygon and the curve. This process allowed him to establish upper and lower bounds for the area or volume in question.134

For example, to determine the value of pi, he used polygons with up to 96 sides to bound the circle's circumference. Similarly, he proved that the area of a parabolic segment is 4/3 of the area of a triangle with the same base and height. While Archimedes also explored a "mechanical" method involving infinitesimally thin strips to discover these results, he relied exclusively on the rigorous method of exhaustion for his formal mathematical proofs.45