A272487 Decimal expansion of the edge length of a regular heptagon with unit circumradius.

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

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 = 7, and the constant, a = e(7), is the smallest m for which e(m) 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

			0.8677674782351162409515366656967175092199814555749197528890946...
		

Links

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

Crossrefs

Cf. A004169, A019434.

Cf. A160389.

Edge lengths of nonconstructible n-gons: A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

Programs

Mathematica

N[2*Sin[Pi/7], 25] (* G. C. Greubel, May 01 2016 *)
RealDigits[2*Sin[Pi/7],10,120][[1]] (* Harvey P. Dale, Mar 07 2020 *)

PARI

2*sin(Pi/7)

Formula

Equals 2*sin(Pi/7) = 2*cos(Pi*5/14).

Equals i^(-5/7) + i^(5/7). - Gary W. Adamson, Feb 12 2022

One of the 6 real-valued roots of x^6 -7*x^4 +14*x^2 -7 =0. - R. J. Mathar, Aug 29 2025