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A258146
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Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.
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5
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1, 8, 1, 6, 9, 0, 1, 1, 3, 8, 1, 6, 2, 0, 9, 3, 2, 8, 4, 6, 2, 2, 3, 2, 4, 7, 3, 2, 5, 4, 9, 7, 1, 2, 7, 5, 9, 3, 1, 0, 8, 0, 7, 0, 8, 5, 1, 9, 0, 8, 7, 1, 0, 2, 5, 0, 4, 6, 6, 5, 3, 1, 1, 8, 8, 2, 2, 0, 6, 4, 0, 4, 7, 3, 1, 5, 4, 6, 9, 2, 9, 8, 1, 9, 7, 7, 2, 3, 9, 4, 4, 6, 7, 4, 9, 3, 8, 2, 8, 0, 8
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Area ratio ar = (1 - 2/Pi)/2 = 0.181690113816209...
For Buffon's constant 2/Pi see A060294.
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MATHEMATICA
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RealDigits[(1-2/Pi)/2, 10, 120][[1]] (* Harvey P. Dale, Sep 23 2017 *)
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KEYWORD
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AUTHOR
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STATUS
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approved
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