A104956 -id:A104956 - cp's OEIS frontend

A010527 Decimal expansion of sqrt(3)/2.

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

Offset: 0

N. J. A. Sloane

This is the ratio of the height of an equilateral triangle to its base.
Essentially the same sequence arises from decimal expansion of square root of 75, which is 8.6602540378443864676372317...
Also the real part of i^(1/3), the cubic root of i. - Stanislav Sykora, Apr 25 2012
Gilbert & Pollak conjectured that this is the Steiner ratio rho_2, the least upper bound of the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 2. (See Ivanov & Tuzhilin for the status of this conjecture as of 2012.) - Charles R Greathouse IV, Dec 11 2012
Surface area of a regular icosahedron with unit edge is 5*sqrt(3), i.e., 10 times this constant. - Stanislav Sykora, Nov 29 2013
Circumscribed sphere radius for a cube with unit edges. - Stanislav Sykora, Feb 10 2014
Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - Enrique Pérez Herrero, Nov 13 2014
Area of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016
If a, b, c are the sides of a triangle ABC and h_a, h_b, h_c the corresponding altitudes, then (h_a+h_b+h_c) / (a+b+c) <= sqrt(3)/2; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 26 2022

			0.86602540378443864676372317...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.2, 8.3 and 8.6, pp. 484, 489, and 504.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), pp. 450-451.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, and J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.8, page 114.

Harry J. Smith, Table of n, a(n) for n = 0..20000
E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16, (1968), pp. 1-29.
A. O. Ivanov and A. A. Tuzhilin, The Steiner ratio Gilbert-Pollak conjecture is still open, Algorithmica 62:1-2 (2012), pp. 630-632.
Matt Parker, The mystery of 0.866025403784438646763723170752936183471402626905190314027903489, Stand-up Maths, YouTube video, Feb 14 2024.
Simon Plouffe, Plouffe's Inverter, sqrt(3)/2 to 10000 digits.
Simon Plouffe, Sqrt(3)/2 to 5000 digits.
Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem.
Wikipedia, Icosahedron.
Wikipedia, Platonic solid.
Wikipedia, Unified Thread Standard.
Index entries for algebraic numbers, degree 2.

Cf. A010153.
Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron).
Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Cf. A126664 (continued fraction), A144535/A144536 (convergents).
Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).

SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[Sqrt[3]/2, 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=10*(sqrt(3)/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010527.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

sqrt(3)/2 \\ Michel Marcus, Apr 10 2016

Equals cos(30 degrees). - Kausthub Gudipati, Aug 15 2011
Equals A002194/2. - Stanislav Sykora, Nov 30 2013
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals 1/(10*A020832). - Bernard Schott, Sep 29 2022
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
Equals 2F1(-1/4,1/4 ; 1/2 ; 3/4) . - R. J. Mathar, Aug 31 2025

Last term corrected and more terms added by Harry J. Smith, Jun 02 2009

A120011 Decimal expansion of sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Eric Desbiaux, Jul 04 2008

Area of equilateral triangle of side 1.
Quadratic number with denominator 4 and minimal polynomial 16x^2 - 3. - Charles R Greathouse IV, Jun 30 2021
With offset 1, surface area of a pentagonal bipyramid (Johnson solid J_13) with unit edges. - Paolo Xausa, Aug 04 2025

			0.43301270189221932338186158537646809173570131345259515701395....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Pentagonal bipyramid.
Index entries for algebraic numbers, degree 2

Cf. A010527.

Cf. Areas of higher regular polygons: A102771, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[Sqrt[3]/4,10,120][[1]] (* Harvey P. Dale, Feb 18 2016 *)

sqrt(3)/4 \\ Charles R Greathouse IV, Jun 26 2013

A090488 Decimal expansion of 2 + 2*sqrt(2).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A010482 Decimal expansion of square root of 27.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {5, 10} repeated (A040021). - Harry J. Smith, Jun 04 2009
6 + sqrt(27) represents the surface of a dodecahedron of side equal to one. S = 3*a^2(2 + sqrt(3)) with a = 1. - Vincenzo Librandi, Jul 10 2010
sqrt(27) is the perimeter of an equilateral triangle whose incircle's diameter is 1. - Martin Janecke, May 31 2016
If r = 2*a * sin(3t)/sin(2t) and x*(x^2+y^2) = a * (3x^2-y^2) are respectively a polar equation and a Cartesian equation of the Maclaurin trisectrix, then sqrt(27) * a^2 = area of the loop of this trisectrix = area between the curve and its asymptote (see Mathcurve link). - Bernard Schott, Jul 14 2020
Area of a regular hexagon with side length sqrt(2). - Christoph B. Kassir, Sep 29 2022
The solution of x^sqrt(3)=sqrt(3)^x, see e.g. A360148. - R. J. Mathar, Mar 24 2023
Surface area of a snub disphenoid (Johnson solid J_84) with unit edges. - Paolo Xausa, Aug 02 2025

			5.196152422706631880582339024517617100828415761431141884167420938355799....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Robert Ferréol, MacLaurin trisectrix, Mathcurve.
Wikipedia, Snub disphenoid.
Index to sequences related to curves.
Index entries for algebraic numbers, degree 2

Cf. A040021 (continued fraction), A248254 (Egyptian fraction).

Cf. A104956 (half), A002194 (sqrt(3)).

RealDigits[N[Sqrt[27], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(27); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010482.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Equals 3*sqrt(3) = 3 * A002194. - Bernard Schott, Jul 14 2020

Equals 2 * A104956. - Christoph B. Kassir, Oct 02 2022

A102771 Decimal expansion of area of a regular pentagon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

			1.720477400588966922759011977...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. Areas of other regular polygons: A120011, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[(5/4)*Sqrt[GoldenRatio^3/Sqrt[5]], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

5/(4*tan(Pi/5)) \\ Michel Marcus, Mar 25 2015

Equals sqrt(25 + 10*sqrt(5)) / 4.
Equals (3*phi+1)*sqrt(3-phi) with the golden section phi = (1 + sqrt(5))/2. - Wolfdieter Lang, Jan 25 2013
Equals 5/(4*tan(Pi/5)). - Michel Marcus, Mar 25 2015
Equals (5/4)*sqrt(phi^3/sqrt(5)). - G. C. Greubel, Jul 03 2017

Corrected the title. - Stanislav Sykora, Apr 12 2015

A010502 Decimal expansion of square root of 48.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

sqrt(48)/10 is the area enclosed by Koch's fractal snowflake based on unit-sided equilateral triangle (actually 8/5 times the latter's area). - Lekraj Beedassy, Jan 06 2005
7+sqrt(48) is the ratio of outer to inner Soddy circles' radii for three identical kissing circles (see Soddy circles link). - Lekraj Beedassy, Feb 14 2006
Continued fraction expansion is 6 followed by {1, 12} repeated. - Harry J. Smith, Jun 06 2009
Let a, b, c the sides of a triangle ABC of area S, then 4*sqrt(3) <= (a^2+b^2+c^2) / S; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 27 2022
Surface area of a gyroelongated square bipyramid (Johnson solid J_17) with unit edges. - Paolo Xausa, Aug 02 2025

			6.928203230275509174109785366023489467771221015241522512223227917807732...

J. N. Kapur, Mathematics Enjoyment For The Millions, Problem 47 pp. 64-67, Arya Book Depot, New Delhi 2000.
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 6.3, page 112.

Harry J. Smith, Table of n, a(n) for n = 1..20000
L. Riddle, Area of the Koch Snowflake
Bernard Schott, Soddy circles
Wikipedia, Gyroelongated square bipyramid.
Index entries for algebraic numbers, degree 2

Cf. A040041 (continued fraction).

Cf. A002194, A104956, A010527, A152623 (other geometric inequalities).

RealDigits[N[Sqrt[48],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)

default(realprecision, 20080); x=sqrt(48); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010502.txt", n, " ", d));  \\ Harry J. Smith, Jun 06 2009

Equals 4*A002194. - R. J. Mathar, Jul 31 2010
Equals A176053/A246724 - 7 (2nd comment and Soddy link). - Bernard Schott, Mar 17 2022
Equals 1/A020805. - Bernard Schott, Sep 28 2022

A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Surface area of a regular hexagonal prism with unit side length and height. - Wesley Ivan Hurt, May 04 2021

			11.196152422706631880582339024517617100828415761431141884167420938355...

G. C. Greubel, Table of n, a(n) for n = 2..10000
Index entries for algebraic numbers, degree 2

Cf. A010482, A090488, A104956, A120011, A176532.

Mathematica
```
RealDigits[N[6+3*Sqrt[3],150]][[1]]
```

sqrt(27)+6 \\ Charles R Greathouse IV, Apr 25 2016

Equals 6+3*sqrt(3).

Equals 1 + A176532 = 6 + A010482. - R. J. Mathar, Jun 25 2010

Offset corrected and keyword:cons inserted by R. J. Mathar, Jun 25 2010

A178817 Decimal expansion of the area of the regular 7-gon (heptagon) of edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

			3.63391244400158899253619300760022057873501036159544491714598040951029...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Heptagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A178818, A343058.

Cf. Areas of other regular polygons: A120011, A102771, A104956, A090488, A256853, A178816, A256854, A178809.

SetDefaultRealField(RealField(100)); R:=RealField(); 7*Cot(Pi(R)/7)/4; // G. C. Greubel, Jan 22 2019

evalf[120]((7/4)*cot(Pi/7)); # Muniru A Asiru, Jan 22 2019

Mathematica
```
RealDigits[7*Cot[Pi/7]/4, 10, 100][[1]]
```

p=7; a=(p/4)*cotan(Pi/p)  \\ Set realprecision in excess. - Stanislav Sykora, Apr 12 2015

numerical_approx(7*cot(pi/7)/4, digits=100) # G. C. Greubel, Jan 22 2019

Equals (7/4) * cot(Pi/7).
From Michal Paulovic, Dec 27 2022: (Start)
Equals 7 / (4 * tan(Pi/7)) = 7 / (4 * A343058).
Equals sqrt(7/3 * (35 + 2 * 196^(1/3) * ((13 - 3 * sqrt(3) * i)^(1/3) + (13 + 3 * sqrt(3) * i)^(1/3)))) / 4.
Equals sqrt(7/4) * sqrt(35/12 + (637/54 - sqrt(-2401/108))^(1/3) + (637/54 + sqrt(-2401/108))^(1/3)).
(End)
A root of the polynomial 4096*x^6 - 62720*x^4 + 115248*x^2 - 16807. - Joerg Arndt, Jan 02 2023

A256853 Decimal expansion of the area of a unit 9-gon.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 12 2015

From Michal Paulovic, May 09 2024: (Start)
This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.
9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

			6.181824193772900127213744059619763614941713348134358098386864...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Nonagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024

RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 9.
From Michal Paulovic, May 09 2024: (Start)
Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.
Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.
Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)
Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025

A178816 Decimal expansion of the area of the regular 10-gon (decagon) of edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

An algebraic number with degree 4 and denominator 2; minimal polynomial 16x^4 - 1000x^2 + 3125. - Charles R Greathouse IV, Apr 25 2016

This equals in a regular pentagon inscribed in a unit circle with vertices V0 = (x, y) = (1, 0), and V1..V4 in the counterclockwise sense, one tenth of the y-coordinate of the midpoint of side (V1,V2), named M1: M1_y = (2*sqrt(3 - phi) + sqrt(7 - 4*phi))/4 = sqrt(3 + 4*phi)/4. The x-coordinate is M1_x = -1/4. - Wolfdieter Lang, Jan 09 2018

			7.69420884293813350642572644009227456001675535884448106759789062593715...
sqrt(3 + 4*phi)/4 = 0.769420884293813350642572644009227456001675535884... - _Wolfdieter Lang_, Jan 09 2018

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Wikipedia, Decagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 4

Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A090488, A256853, A256854, A178809.

Cf. A001622.

SetDefaultRealField(RealField(100)); 5*Sqrt(2*Sqrt(5)+5)/2; // G. C. Greubel, Jan 22 2019

evalf[120](5*sqrt(5+2*sqrt(5))/2); # Muniru A Asiru, Jan 22 2019

RealDigits[5*Sqrt[5+2*Sqrt[5]]/2, 10, 100][[1]]

5*sqrt(2*sqrt(5)+5)/2 \\ Charles R Greathouse IV, Apr 25 2016

numerical_approx(5*sqrt(2*sqrt(5)+5)/2, digits=100) # G. C. Greubel, Jan 22 2019

Digits of 5*sqrt(5+2*sqrt(5))/2 = (5/2)*sqrt(3 + 4*phi), with phi from A001622.

cp's OEIS Frontend

A010527 Decimal expansion of sqrt(3)/2.

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A120011 Decimal expansion of sqrt(3)/4.

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A090488 Decimal expansion of 2 + 2*sqrt(2).

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A010482 Decimal expansion of square root of 27.

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A102771 Decimal expansion of area of a regular pentagon with unit edge length.

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A010502 Decimal expansion of square root of 48.

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A178809 Decimal expansion of the area of the regular 12-gon (dodecagon) of edge length 1.