A385847 Decimal expansion of the length of the chord in a circle (in units of the diameter) that splits the area into one and two thirds.

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

Offset

0,1

Comments

A chord length c and the area A of the circular segment are related by A = Pi^2*[arcsin x - x *sqrt(1-x^2)] where x = c/(2r) is the chord length divided by diameter 2r.

Setting A = Pi^2*r/3 yields this constant here. Setting A=Pi^2*r/2 gives x=1: a chord through the circle center splits the area into two halves.

arccos(x) = 0.268133... rad = 15.36291.. deg is the angle between the chord and the line through the circle center (that cuts it in half) measured at the point where the chord and the line intersect the circle.

Links

Table of n, a(n) for n=0..112.

Formula

Root of arcsin(x) -x * sqrt(1-x^2) = Pi/3.

Example

0.96426707428389728406 ... = 1.928534148567794... / 2 .

Mathematica

RealDigits[Chop[x /. FindRoot[ArcSin[x] - x*Sqrt[1 - x^2] - Pi/3, {x, 1}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, Apr 17 2026 *)

Crossrefs

Cf. A192408 (for right hand side Pi/6, splitting area 1:5).

Sequence in context: A331550 A253267 A388562 * A309070 A010544 A343308

Adjacent sequences: A385844 A385845 A385846 * A385848 A385849 A385850

Keyword

cons,nonn

Author

R. J. Mathar, Oct 01 2025

Status

approved