OFFSET
0,1
COMMENTS
Given a unit circle and a square of equal area, what is the amount of the square peg shavings (or filings) which would allow the peg to be inserted into the circle? It turns out to be not quite two sevenths.
REFERENCES
Daniel Zwillinger, Editor, CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, Section 4.6.6 Circles, page 334 & figure 4.18, 2003.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
1728 Software Systems, Circle Sector, Segment, Chord and Arc Calculator.
Eric Weisstein's World of Mathematics, Circular Segment.
Wikipedia, Square peg in a round hole.
Wikipedia, Squaring the circle.
Wikipedia, Diagram of the problem.
FORMULA
Area = 4*arccos(sqrt(Pi)/2) - sqrt(Pi*(4-Pi)).
Area = Pi + sqrt(2*Pi(2 - sqrt(Pi*(4 - Pi)))) - 4*arcsin(sqrt(Pi/4)). - Robert G. Wilson v, Mar 19 2014
EXAMPLE
0.28445855040980187815920101812693174533005283078946269804587750...
MATHEMATICA
RealDigits[ 4*ArcCos[ Sqrt[Pi]/2] - Sqrt[ Pi(4 - Pi)], 10, 111][[1]]
RealDigits[Pi + Sqrt[ 2Pi(2 - Sqrt[Pi (4 - Pi)])] - 4 ArcSin[ Sqrt[Pi/4]], 10, 111][[1]] (* Robert G. Wilson v, Sep 20 2011 *)
PROG
(PARI) 4*acos(sqrt(Pi)/2) - sqrt(Pi*(4-Pi)) \\ G. C. Greubel, Mar 28 2017
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
William H. Richardson and Robert G. Wilson v, Sep 05 2011
STATUS
approved