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

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Infinity — infinity Mathematical symbol representing the concept of infinity.

The method of exhaustion was a rigorous logical technique developed by Eudoxus of Cnidus to calculate the areas and volumes of geometric figures by approximating them with simpler shapes of known dimensions, such as polygons or prisms.123

Rather than using limits or infinitesimal quantities, Eudoxus employed a strictly logical procedure based on the theory of proportions. To calculate an area, he would inscribe a sequence of polygons within a figure; as the number of sides increased, the polygons progressively occupied more of the figure's area.14523

Eudoxus proved his results by showing that any value less than the target area could be exceeded by an inscribed polygon, and any value greater could be undercut by a circumscribed one. By demonstrating that all other possibilities were "exhausted," he established the exact area. For example, he used this method to prove that the area of a circular disk is proportional to the square of its radius and that the volume of a pyramid or cone is one-third that of a prism or cylinder with the same base and height.25