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

Author

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

Keywords

• cons
• nonn

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.

Examples

			0.28445855040980187815920101812693174533005283078946269804587750...
		

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.

Crossrefs

Cf. A000796, A019704, A127454.

Programs

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 *)

PARI

4*acos(sqrt(Pi)/2) - sqrt(Pi*(4-Pi)) \\ G. C. Greubel, Mar 28 2017

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