OFFSET
0,2
COMMENTS
The formula for the ratio of the area of a circular segment with central angle alpha and the area of one half of the corresponding circular disk is (alpha - sin(alpha))/Pi. Here alpha = Pi/2.
This is also the ratio of the area of a circular disk without a central inscribed rectangle (2*x, 2*y) together with the two opposite circular segments each with central angle beta and the area of the circular disk. This is the analog of the ratio of the volume of a sphere with missing central cylinder symmetric hole of length 2*y and the area of the sphere. See a comment on A019699. In two dimensions this problem is not remarkable, because the radius R of the circle does matter. The formula is here: area ratio ar = 1 - (beta + sin(beta)/Pi) where beta = arcsin(2*yhat*sqrt(1-yhat^2)), with yhat = y/R, and beta = Pi - alpha from above.
The astonishing result from three dimensions, ar_3 = yhat^3, could suggest ar = yhat^2, which is wrong. Thanks to Sven Heinemeyer for inspiring me to look into this.
Essentially the same digit sequence as A188340. - R. J. Mathar, Jun 12 2015
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Circular Segment.
FORMULA
Area ratio ar = (1 - 2/Pi)/2 = 0.181690113816209...
For Buffon's constant 2/Pi see A060294.
MATHEMATICA
RealDigits[(1-2/Pi)/2, 10, 120][[1]] (* Harvey P. Dale, Sep 23 2017 *)
PROG
(PARI) 1/2 - 1/Pi \\ Michel Marcus, Oct 19 2017
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 31 2015
STATUS
approved