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Archimedes' Recurrence Formula


ArchimedesRecurrence

Let a_n and b_n be the perimeters of the circumscribed and inscribed n-gon and a_(2n) and b_(2n) the perimeters of the circumscribed and inscribed 2n-gon. Then

a_(2n)=(2a_nb_n)/(a_n+b_n)
(1)
b_(2n)=sqrt(a_(2n)b_n).
(2)

The first follows from the fact that side lengths of the polygons on a circle of radius r=1 are

s_R=2tan(pi/n)
(3)
s_r=2sin(pi/n),
(4)

so

a_n=2ntan(pi/n)
(5)
b_n=2nsin(pi/n).
(6)

But

(2a_nb_n)/(a_n+b_n)=(2·2ntan(pi/n)·2nsin(pi/n))/(2ntan(pi/n)+2nsin(pi/n))
(7)
=4n(tan(pi/n)sin(pi/n))/(tan(pi/n)+sin(pi/n)).
(8)

Using the identity

 tan(1/2x)=(tanxsinx)/(tanx+sinx)
(9)

then gives

 (2a_nb_n)/(a_n+b_n)=4ntan(pi/(2n))=a_(2n).
(10)

The second follows from

 sqrt(a_(2n)b_n)=sqrt(4ntan(pi/(2n))·2nsin(pi/n)).
(11)

Using the identity

 sinx=2sin(1/2x)cos(1/2x)
(12)

gives

sqrt(a_(2n)b_n)=2nsqrt(2tan(pi/(2n))·2sin(pi/(2n))cos(pi/(2n)))
(13)
=4nsqrt(sin^2(pi/(2n)))
(14)
=4nsin(pi/(2n))
(15)
=b_(2n).
(16)

Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi (pi).


See also

Archimedes Algorithm, Pi

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References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, p. 186, 1965.

Referenced on Wolfram|Alpha

Archimedes' Recurrence Formula

Cite this as:

Weisstein, Eric W. "Archimedes' Recurrence Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ArchimedesRecurrenceFormula.html

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