A104955 -id:A104955 - cp's OEIS frontend

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

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

Offset: 1

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

Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.
Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - R. J. Mathar, Jul 26 2007
This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - Jean-François Alcover, Dec 19 2016
Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - Charles R Greathouse IV, May 21 2022
For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022
Surface area of a triangular bipyramid (Johnson solid J_12) with unit edges. - Paolo Xausa, Aug 04 2025

			2.59807621135331594029116951225880855041420788071557094208371046917789952536320...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

G. C. Greubel, Table of n, a(n) for n = 1..10000
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Kiran S. Kedlaya, A < B, (1999), Problem 6.1, p. 6.
J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.
Michael Penn, not as bad as it seems..., YouTube video, 2021.
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Eric Weisstein's World of Mathematics, Twenty-Vertex Entropy Constant.
Wikipedia, Hexagon.
Wikipedia, Regular polygon.
Wikipedia, Triangular bipyramid.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A104954, A104955, A104957.

Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

3*sqrt(3)/2 \\ G. C. Greubel, Jul 03 2017

Equals (3*sqrt(3))/2, that is, 2*A104954.

Equals Product_{k>=3} (((k - 1)^2*(k + 2))/((k + 1)^2*(k - 2)))^(k/2). - Antonio Graciá Llorente, Oct 13 2024

A104954 Decimal expansion of the area of the regular triangle with circumradius 1.

Original entry on oeis.org

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

Offset: 1

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

Equivalently, the area in the complex plane of the smallest convex set containing all order-3 roots of unity.

			1.299038105676657970145584756129404275207103940357785471...

Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319.
Index entries for algebraic numbers, degree 2

Cf. A104955, A104956, A104957.

RealDigits[N[Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 3, 108]][[1]]

Equals 3*sqrt(3)/4.
Equals (1/2)*A104956.
Equals 2F1(1/6,1/3 ; 1/2 ; 25/27). [Zucker] - R. J. Mathar, Jun 24 2024

Edited by Georg Fischer, Jul 29 2021

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

Original entry on oeis.org

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

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

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]

			2.736410188638104330479555843359202125813170815480406351432222353...

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

Cf. A104954, A104955, A104956.

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]]

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

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

A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 29 2015

Quartic number of degree 4 and denominator 2; minimal polynomial 16x^4 - 500x^2 + 3125. - Charles R Greathouse IV, Apr 20 2016

			2.9389262614623656458435297731953638429882621882157299553613624...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein's MathWorld, Decagon
Wikipedia, Decagon

Cf. A104954 (triangle), A104955 (pentagon), A104956 (hexagon), A104957 (heptagon).

Cf. A178816 (area of decagon with edge length 1). A182007.

RealDigits[(5/2)*Sqrt[(5 - Sqrt[5])/2], 10, 101] // First

(5/2)*sqrt((5 - sqrt(5))/2) \\ Michel Marcus, May 29 2015

Equals (5/2)*sqrt((5-sqrt(5))/2).
Area formulas from triangle to dodecagon, with circumradius 1:
   n-gon    area(n) = (1/2)*n*sin(2*Pi/n)
   3-gon    (3*sqrt(3))/4
   4-gon    2
   5-gon    (5/4)*sqrt((5+sqrt(5))/2)
   6-gon    (3*sqrt(3))/2
   7-gon    (7/2)*cos((3*Pi)/14)
   8-gon    2*sqrt(2)
   9-gon    (9/2)*sin((2*Pi)/9)
  10-gon    (5/2)*sqrt((5-sqrt(5))/2)
  11-gon    (11/2)*sin((2*Pi)/11)
  12-gon    3
This constant is (5/2)*A182007. - Wolfdieter Lang, May 08 2018

A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 12 2021

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

			1.12256994144896343110486287949381696894803120580270...

Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
Mathematics Stack Exchange, Area of a five pointed star, 2014.
Wikipedia, Pentagram.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).
Equals 5/(cot(Pi/5) + cot(Pi/10)).
Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).
Equals (5/2)*sqrt((25 -11*sqrt(5))/2).
Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

cp's OEIS Frontend

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

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A104954 Decimal expansion of the area of the regular triangle with circumradius 1.

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

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A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

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A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

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