A231984 Decimal expansion of the solid angle (in deg^2) of a spherical square having sides of one degree.

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

Offset: 0

Author

Stanislav Sykora, Nov 17 2013

Keywords

• nonn
• cons
• easy

Comments

See the comments to A231983 which will make it clear why on a sphere the solid angle of a square with one degree arc-length side is not exactly one deg^2. The correct value, shown here, is A231983*A231981.

Examples

			0.9999746164392786543219850947849682255179591524185764527406467...
		

References

• G. V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Wikipedia, Solid angle, Section 3.3 (Pyramid)
• Wikipedia, Square degree
• Wikipedia, Steradian

Crossrefs

Cf. A000796 (Pi), A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231983 (this constant in sr), A231987 (for square with 1 rad side), A231985, A231986.

Programs

Mathematica

RealDigits[4*ArcSin[Sin[Pi/360]^2](180/Pi)^2,10,120][[1]] (* Harvey P. Dale, Aug 20 2017 *)

PARI

default(realprecision, 120);
4*asin(sin(Pi/360)^2)*(180/Pi)^2 \\ Rick L. Shepherd, Jan 28 2014

Formula

4*arcsin(sin(R/2)sin(S/2))*(180/Pi)^2, where R = S = Pi/180.