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Kochanski's Approximation


KochanskysConstruction

The approximation for pi given by

pi approx sqrt((40)/3-2sqrt(3))
(1)
=1/3sqrt(120-18sqrt(3))
(2)
=3.141533....
(3)

In the above figure, let OA=OF=1, and construct the circle centered at A=(0,0) of radius 1. This intersects O at point B=(-sqrt(3)/2,1/2). Now construct the circle about B with radius 1. The circles A and B intersect in C=(-sqrt(3)/2,-1/2), and the line CO intersects the perpendicular to OA through A in the point D=(-sqrt(3)/3,0). Now construct the point E=(3-sqrt(3)/3,0) to be a distance 3 along DA. The line segment EF is then of length

 sqrt(2^2+(3-1/3sqrt(3))^2)=sqrt((40)/3-2sqrt(3)).
(4)

This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1999).


See also

Geometric Construction, Pi

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References

Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 44, 1982.Kochansky, A. A. "Observationes Cyclometricae ad facilitandam Praxin accomodatae." Acta Eruditorum 4, 394-398, 1685.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 143, 1999.

Referenced on Wolfram|Alpha

Kochanski's Approximation

Cite this as:

Weisstein, Eric W. "Kochanski's Approximation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KochanskisApproximation.html

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