page_title: Viète's formula
section_title: Derivation
hierarchical_section_title: Viète's formula / Derivation
context_page_description: In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant π:
  It is named after François Viète, who published it in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII.
context_section_description: Viète obtained his formula by comparing the areas of regular polygons with 2ⁿ and 2ⁿ ⁺ ¹ sides inscribed in a circle. The first term in the product, , is the ratio of areas of a square and an octagon, the second term is the ratio of areas of an octagon and a hexadecagon, etc. Thus, the product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a 2ⁿ-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon (the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.
  Another derivation is possible based on trigonometric identities and Euler's formula.
  By repeatedly applying the  double-angle formula
  one may prove by mathematical induction that, for all positive integers n,
  The term 2ⁿ sin  goes to x in the limit as n goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution x = .
caption: A sequence of regular polygons with numbers of sides equal to powers of two, inscribed in a circle. The ratios between areas or perimeters of consecutive polygons in the sequence give the terms of Viète's formula.