A272488 Decimal expansion of the edge length of a regular 9-gon with unit circumradius.

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

Offset: 0

Author

Stanislav Sykora, May 01 2016

Keywords

• nonn
• cons
• easy
• changed

Comments

The edge length e(m) of a regular m-gon is e(m) = 2*sin(Pi/m). In this case, m = 9, and the constant, a = e(9), is not constructible using a compass and a straightedge (see A004169). With an odd m, in fact, e(m) would be constructible only if m were a Fermat prime (A019434).

Examples

			0.6840402866513374660881992293645191615261667350283212569300969953...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Wikipedia, Constructible number
• Wikipedia, Regular polygon
• Index entries for algebraic numbers, degree 6

Crossrefs

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

Programs

Mathematica

RealDigits[N[2Sin[Pi/9], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI

2*sin(Pi/9)

Formula

Equals 2*sin(Pi/9) = 2*cos(Pi*7/18) = 2*A019829.

Equals Im((4+4*sqrt(3)*i)^(1/3)). - Gerry Martens, Mar 19 2024

A root of x^6 -6*x^4 +9*x^2 -3 =0. - R. J. Mathar, Aug 29 2025