Proposition XII.2 — Circles are to one another as the squares on the diameters.

Proof

Apply the method of exhaustion: inscribe similar polygons in the two circles; by XII.1 they are in the ratio of squares on the diameters. Any deviation from that ratio at the level of the circles leads, via X.1, to a contradiction by choosing inscribed polygons close enough to fill the circle.

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Depends on (2)

X.1Proposition X.1Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and…
XII.1Proposition XII.1Similar polygons inscribed in circles are to one another as the squares on the diameters.

Required by (dependents) (2)

XII.10Proposition XII.10Any cone is a third part of the cylinder which has the same base with it and equal height.
XII.11Proposition XII.11Cones and cylinders which are of the same height are to one another as their bases.

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