Dividing a circle into areas lines (there are two points on one side and one on the other). So the recurrence can be expressed as which can be easily reduced to using the sums of the first formula_3 natural numbers and the first formula_3 squares, this combines to Finally which yields The lemma asserts that the number of regions is maximal if all "inner" intersections of chords are simple (exactly two chords pass through each point of intersection in the interior). This will be the case if the points on the circle are chosen "in general position". Under this assumption of "generic intersection", the number
Area of a circle approximation among rational numbers with denominator up to 113. The number ⁄ is also an excellent approximation to , better than any other rational number with denominator less than 16604. Snell proposed (and Huygens proved) a tighter bound than Archimedes': This for "n" = 48 gives a better approximation (about 3.14159292) than Archimedes' method for "n" = 768. Let one side of an inscribed regular "n-"gon have length "s" and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle