The most common example of this was Archimedes' calculation of the area of a circle using inscribed polygons
. Archimedes found that by increasing the number of sides of an inscribed polygon
, the area of the polygon became closer to that of the circle.
He proves this by circumscribing a polygon about the circle and inscribing another inside the circle (as in proposition 33 above), then proving that the circumscribed polygon is greater than the triangle ABC and the inscribed polygon
To calculate this area in the classroom, we inscribed polygons
inside the circle, much as Antiphon did.