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Archimedes' Recurrence Formula


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

$\displaystyle a_{2n}$ $\textstyle =$ $\displaystyle {2a_nb_n\over a_n+b_n}$ (1)
$\displaystyle b_{2n}$ $\textstyle =$ $\displaystyle \sqrt{a_{2n}b_n}\,.$ (2)

The first follows from the fact that side lengths of the Polygons on a Circle of Radius $r=1$ are
$\displaystyle s_R$ $\textstyle =$ $\displaystyle 2\tan\left({\pi\over n}\right)$ (3)
$\displaystyle s_r$ $\textstyle =$ $\displaystyle 2\sin\left({\pi\over n}\right),$ (4)

$\displaystyle a_n$ $\textstyle =$ $\displaystyle 2n\tan\left({\pi\over n}\right)$ (5)
$\displaystyle b_n$ $\textstyle =$ $\displaystyle 2n\sin\left({\pi\over n}\right).$ (6)

$\displaystyle {2a_nb_n\over a_n+b_n}$ $\textstyle =$ $\displaystyle {2\cdot 2n\tan\left({\pi\over n}\right)\cdot 2n\sin\left({\pi\over n}\right)\over 2n\tan\left({\pi\over n}\right)+2n\sin\left({\pi\over n}\right)}$  
  $\textstyle =$ $\displaystyle 4n {\tan\left({\pi\over n}\right)\sin\left({\pi\over n}\right)\over\tan\left({\pi\over n}\right)+\sin\left({\pi\over n}\right)}.$ (7)

Using the identity
\tan({\textstyle{1\over 2}}x)={\tan x\sin x\over\tan x+\sin x}
\end{displaymath} (8)

then gives
{2a_nb_n\over a_n+b_n}=4n\tan\left({\pi\over 2n}\right)=a_{2n}.
\end{displaymath} (9)

The second follows from
\sqrt{a_{2n}b_n}=\sqrt{4n\tan\left({\pi\over 2n}\right)\cdot 2n\sin\left({\pi\over n}\right)}
\end{displaymath} (10)

Using the identity
\sin x=2\sin({\textstyle{1\over 2}}x)\cos({\textstyle{1\over 2}}x)
\end{displaymath} (11)

$\sqrt{a_{2n}b_n} = 2n\sqrt{2\tan\left({\pi\over 2n}\right)\cdot 2\sin\left({\pi\over 2n}\right)\cos\left({\pi\over 2n}\right)}$
$ = 4n\sqrt{\sin^2\left({\pi\over 2n}\right)} = 4n\sin\left({\pi\over 2n}\right)=b_{2n}.\quad$ (12)
Successive application gives the Archimedes Algorithm, which can be used to provide successive approximations to Pi ($\pi$).

See also Archimedes Algorithm, Pi


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

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© 1996-9 Eric W. Weisstein