Measurement of a Circle

Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion.

He found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons.

[6] However, these bounds are familiar from the study of Pell's equation and the convergents of an associated simple continued fraction, leading to much speculation as to how much of this number theory might have been accessible to Archimedes.

Discussion of this approach goes back at least to Thomas Fantet de Lagny, FRS (compare Chronology of computation of π) in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen.

But the bounds can also be produced by an iterative geometrical construction suggested by Archimedes' Stomachion in the setting of the regular dodecagon.