A104957 Decimal expansion of the area of the regular 7-gon (heptagon) of circumradius = 1.

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

Offset: 1

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Keywords

• nonn
• cons
• changed

Comments

Equivalently, the area in the complex plane of the smallest convex set containing all order-7 roots of unity.

The second largest root of 4096*x^6 - 87808*x^4 + 537824*x^2 - 823543 = 0. [Corrected by Sean A. Irvine, May 24 2025]

Examples

			2.736410188638104330479555843359202125813170815480406351432222353...
		

Links

• Eric Weisstein et al., Root of Unity.
• Eric Weisstein's World of Mathematics, de Moivre Number.
• Index entries for algebraic numbers, degree 6.

Crossrefs

Cf. A104954, A104955, A104956.

Programs

Mathematica

RealDigits[Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 7, 10, 100][[1]]
RealDigits[Root[-823543 + 537824*#1^2 - 87808*#1^4 + 4096*#1^6 &, 5, 0], 10, 100][[1]]

PARI

3*sin((2*Pi)/7) - sin((12*Pi)/7)/2 \\ Michel Marcus, Feb 25 2023

Formula

Equals 3*sin(2*Pi/7) - sin(12*Pi/7)/2.

Equals 7*cos(3*Pi/14)/2. - Amiram Eldar, Feb 25 2023

A root of 4096*x^6 -87808*x^4 +537824*x^2 -823543=0. - R. J. Mathar, Aug 29 2025