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Why did Archimedes use inscribed and circumscribed polygons to approximate pi?

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Portrait of Archimedes by Giuseppe Nogari shows an elderly man with white hair and beard, holding a compass, gazing downward, set against a dark background. — Archimedes Archimedes, oil on canvas by Giuseppe Nogari, 18th century; in the Pushkin Fine Arts Museum, Moscow. Archimedes was an ancient Greek mathematician and inventor.

Archimedes used inscribed and circumscribed polygons to approximate pi as part of the method of exhaustion, a technique used to study difficult-to-measure curved shapes by using simpler, known geometric figures.12

By placing a circle between an inscribed polygon (inside the circle) and a circumscribed polygon (outside the circle), Archimedes could bound the circle's circumference between the calculable perimeters of these polygons. As the number of sides of these polygons increases, they "exhaust" the space of the circle, allowing for increasingly precise measurements.12

Using 96-sided polygons, Archimedes established that the value of pi lies between the lower bound of $3 \frac{10}{71}$ and the upper bound of $3 \frac{1}{7}$ . This approach allowed him to move beyond simple approximation to define rigorous mathematical bounds for the ratio of a circle's circumference to its diameter.12