A256853 Decimal expansion of the area of a unit 9-gon.

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

Offset: 1

Author

Stanislav Sykora, Apr 12 2015

Keywords

• nonn
• cons
• changed

Comments

From Michal Paulovic, May 09 2024: (Start)

This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.

9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

Examples

			6.181824193772900127213744059619763614941713348134358098386864...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Wikipedia, Nonagon
• Wikipedia, Regular polygon
• Index entries for algebraic numbers, degree 6

Crossrefs

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

Programs

Maple

evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024

Mathematica

RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

PARI

p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Formula

Equals (p/4)*cot(Pi/p), with p = 9.

From Michal Paulovic, May 09 2024: (Start)

Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.

Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.

Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.

Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)

Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025