How did Euclid use the method of exhaustion to compare circles and spheres?

In the Elements, Euclid applied the method of exhaustion—a technique originally developed by Eudoxus of Cnidus—to rigorously prove propositions regarding the properties of curved figures in Book XII.12

Rather than using modern concepts like limits or infinitesimals, the method relies on a strictly logical procedure that approximates a curved figure by using a sequence of rectilinear figures (such as polygons or polyhedra) whose areas or volumes are calculable. By increasing the number of sides or faces of these inscribed figures, the difference between the curved figure and the rectilinear approximation is progressively reduced, or "exhausted," until the remaining area or volume is smaller than any assignable quantity.324

Using this approach, Euclid proved that:
* The areas of circles are to one another as the squares of their diameters.12
* The volumes of spheres are to one another as the cubes of their diameters.12

This method allowed Euclid to establish precise geometric relationships for curved shapes by grounding them in the properties of simpler, known rectilinear figures.52