A233700 Decimal expansion of 1/sin(arctan(1/t)) or t/sin(arctan(t)) where t = 2*Pi: hypotenuse for a right triangle of equal area to a disk.

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

Offset: 1

Author

John W. Nicholson, Dec 16 2013

Keywords

• nonn
• cons
• nice

Comments

"The great mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle." (Quote from Wikipedia link)

Examples

			6.362265131567328393691245440586804410699714985138989686582041617045998587331...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Wikipedia, Area of a disk: Triangle method

Crossrefs

Cf. A233527, A019692, A233528.

Programs

Julia

using Nemo
RR = RealField(310)
t = const_pi(RR) + const_pi(RR)
t/sin(atan(t)) |> println # Peter Luschny, Mar 13 2018

Magma

C := ComplexField(); Sqrt(1 + 4*Pi(C)^2) // G. C. Greubel, Jan 08 2018

Magma

R:=RealField(110); SetDefaultRealField(R); n:=Sqrt(1+4*Pi(R)^2); Reverse(Intseq(Floor(10^108*n))); // Bruno Berselli, Mar 13 2018

Mathematica

RealDigits[(2*Pi)/Sin[ArcTan[2*Pi]],10,120][[1]] (* Harvey P. Dale, Jul 12 2014 *)
RealDigits[ Sqrt[1 + 4*Pi^2], 10, 111][[1]] (* Robert G. Wilson v, Mar 12 2015 *)

PARI

sqrt(1+(2*Pi)^2)

Formula

Equals sqrt(1+(2*Pi)^2) = sqrt(1 + (A019692)^2) = sqrt(1 + A212002) = 1/sin(A233527) = A019692/sin(A233528) = 1/cos(A233528) = A019692/cos(A233527).