A194940 The Square Peg in the Round Hole constant.

2, 8, 4, 4, 5, 8, 5, 5, 0, 4, 0, 9, 8, 0, 1, 8, 7, 8, 1, 5, 9, 2, 0, 1, 0, 1, 8, 1, 2, 6, 9, 3, 1, 7, 4, 5, 3, 3, 0, 0, 5, 2, 8, 3, 0, 7, 8, 9, 4, 6, 2, 6, 9, 8, 0, 4, 5, 8, 7, 7, 5, 0, 0, 3, 0, 1, 1, 8, 9, 8, 9, 5, 8, 4, 8, 2, 9, 2, 3, 9, 7, 5, 3, 8, 6, 9, 4, 7, 2, 3, 6, 0, 6, 2, 2, 7, 2, 2, 1, 4, 6, 7, 6, 4, 6, 1, 7, 2, 4, 4, 7

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

Cf. A000796, A019704, A127454.

Sequence in context: A254446 A197486 A086311 * A195920 A227334 A105351

Adjacent sequences: A194937 A194938 A194939 * A194941 A194942 A194943

Keyword

cons,nonn

Author

William H. Richardson and Robert G. Wilson v, Sep 05 2011

Status

approved