A104954 -id:A104954 - 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

A104955 Decimal expansion of the area of the regular 5-gon (pentagon) of circumradius = 1.

Original entry on oeis.org

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

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-5 roots of unity.

			2.377641290737883930291098333448455358514246585314375556118264111075382925212...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Index entries for algebraic numbers, degree 4.

Cf. A104954, A104956, A104957.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 5
RealDigits[(5(Sqrt[(5+Sqrt[5])/2]))/4,10,120][[1]] (* Harvey P. Dale, Jul 21 2013 *)

(5*sqrt((5 + sqrt(5))/2))/4 \\ Michel Marcus, Feb 24 2023

Equals (5*sqrt((5 + sqrt(5))/2))/4.

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

A348637 Largest clock triangle area.

Original entry on oeis.org

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

Offset: 1

Robert B Fowler, Oct 26 2021

Consider an analog clock face to be a unit circle, with unit-length clock hands; the endpoints of the hands lie on the unit circle and form the vertices of a Clock Triangle inscribed within the circle.
The area within this Clock Triangle has maximum value 1.2990353071..., which occurs around 02:54:35 and at its mirror image around 09:05:25.
At time T seconds after 00:00:00, the clock hands are at angles
      S (seconds hand) = T/60 * 360,      (degrees)
      M (minutes hand) = T/60/60 * 360,
      H (hours hand) = T/60/60/12 * 360.
The clock cycle repeats every 12 hours = 43200 seconds.
The second 6 hours of the cycle is a mirror image of the first 6 hours.
The area within the Clock Triangle at any time is equal to
      F(T) = abs(sin(H-M) + sin(M-S) + sin(S-H))/2.
(The derivation of this equation is not overly-complicated.)
The hour and minute hands are exactly 120 degrees apart at times
      T = 14400/11*(3k+1)   and   T = 14400/11*(3k+2)  for integer k.
There are 22 such times during every 12-hour cycle.
Empirically examining the relative extrema of F(T) near these 22 times, it is found that the largest F(T) occurs near T = 10475 (02:54:35), and near its mirror image T = 32725 (09:05:25).
Using Newton's iterative method to solve for Tmax in F'(Tmax) = 0,
      Tmax = 10474.561690797181984...
      F(Tmax) = 1.299035307107332...
Note: an equilateral triangle has area sqrt(3)*3/4 = 1.2990381056...

H. E. Dudeney, Amusements in Mathematics, Dover, 1958, pages 11 and 154; Problem #63, "The Stop-Watch", notes that at times 02:54:35 and 09:05:25 the clock hands are "nearly equidistant" and that "exact equidistance for the three hands is not possible", but does not point out that these two times are the most nearly equidistant times in the 12-hour clock cycle. Of the 430 puzzles in the book, 4 are represented on the front cover illustration; the stop-watch appears in the very center.

Henry Ernest Dudeney, Amusements in Mathematics, London, New York, Nelson, 1917.
Henry Ernest Dudeney, Cover illustration of Amusements in Mathematics with clock, New York, Dover, 1958.

Cf. A104954, A347040, A350141.

f[x_] := (Abs[Sin[2*Pi*x*(1/43200 - 1/3600)] + Sin[2*Pi*x*(1/3600 - 1/60)] + Sin[2*Pi*x*(1/60 - 1/43200)]])/2; RealDigits[FindMaximum[f[x], {x, 10475}, WorkingPrecision -> 110][[1]], 10, 100][[1]] (* Amiram Eldar, Oct 27 2021 *)

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

A333322 Decimal expansion of (3/8) * sqrt(3).

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, Mar 15 2020

This is the area of the regular hexagon of diameter 1.
From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)
For any triangle ABC, where (A,B,C) are the angles:
  sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],
  cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],
and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:
  (ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).
The equalities are obtained only when triangle ABC is equilateral. (End)

			0.649519052838328985...

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

Equals A104954/2 or A104956/4.

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

cp's OEIS Frontend

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

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A104955 Decimal expansion of the area of the regular 5-gon (pentagon) of 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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A348637 Largest clock triangle area.

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

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A333322 Decimal expansion of (3/8) * sqrt(3).

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A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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