cp's OEIS Frontend

A179378 Decimal expansion of the ratio of the area of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r^2 itself.

%I A179378 #15 Aug 19 2025 20:54:27
%S A179378 2,7,7,0,9,7,9,7,6,4,1,8,5,2,1,5,1,8,9,1,4,8,3,3,0,8,6,8,9,5,9,3,8,9,
%T A179378 6,8,0,5,7,8,7,4,5,8,5,7,0,5,5,2,6,2,1,9,0,7,0,2,8,3,1,8,2,1,5,1,0,1,
%U A179378 1,3,1,3,4,4,6,6,1,8,2,2,9,7,9,4,2,5,0,2,8,2,8,5,1,0,5,7,2,5,3,5,2,2,7,2,1
%N A179378 Decimal expansion of the ratio of the area of the triangle corresponding to a circular segment with area r^2 of a circle with radius r to r^2 itself.
%C A179378 In other words, the triangle area is A179378*(r^2). The triangle height is A179377*r. The segment height ("cap height" in MathWorld link) is A179376*r. The chord length is A179375*r. The arc length of the circular segment/sector is r*A179373. The area of the circular segment, r^2, is 1/Pi (A049541) times the area of the circle. The area of the sector is (r^2)*(A179373/2) = (r^2)*(1 + A179378). See references and cross-references for other relationships.
%C A179378 From _Clark Kimberling_, Oct 10 2011: (Start)
%C A179378 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+1) orthogonally; specifically:
%C A179378 xo=0.277097976418521518914833086895...
%C A179378 yo=0.289494183027862650094360757305...
%C A179378 m=1.0447358251025919644670467125044...
%C A179378 |OP|=0.4007370341535820008719293563...  See the Mathematica program for a graph. (End)
%D A179378 S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 7.
%H A179378 G. C. Greubel, <a href="/A179378/b179378.txt">Table of n, a(n) for n = 0..10000</a>
%H A179378 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircularSegment.html">Circular Segment</a>.
%F A179378 Equals sin(A179373)/2 = sin(A179373/2)*cos(A179373/2) = A179375*A179377/2.
%e A179378 .2770979764185215189148330868959389680578745857055262190702831821510113134466...
%t A179378 c = 1;
%t A179378 xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
%t A179378 RealDigits[xo] (* A179378 *)
%t A179378 m = 1/Sin[xo + c]
%t A179378 RealDigits[m]  (* A197009 *)
%t A179378 yo = m*xo
%t A179378 d = Sqrt[xo^2 + yo^2]
%t A179378 Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
%t A179378 ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
%t A179378 AspectRatio -> Automatic, AxesOrigin -> Automatic]
%o A179378 (PARI) sin(solve(x=0, Pi, x-sin(x)-2))/2
%Y A179378 Cf. A179373 (central angle, radians), A179374 (central angle, degrees), A179375 (for chord length), A179376 (for "cap height", height of segment), A179377 (for triangle height), A049541; A197009 (radius of orthogonal circle).
%K A179378 cons,nonn
%O A179378 0,1
%A A179378 _Rick L. Shepherd_, Jul 11 2010