A010527 -id:A010527 - cp's OEIS frontend

A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

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

Offset: 2

Omar E. Pol, Aug 30 2007

Surface area of a regular dodecahedron: A = 3*(sqrt(25 + 10*sqrt(5)))* a^2, where 'a' is the edge.

			20.64572880706760307310814372866331519288849004012237995...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

G. C. Greubel, Table of n, a(n) for n = 2..10001
Eric Weisstein's World of Mathematics, Dodecahedron.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A102769, A001622 (phi), A182007 (associate of phi), A010527 (icosahedron/10), A010469 (octahedron), A002194 (tetrahedron). - Stanislav Sykora, Nov 30 2013

SetDefaultRealField(RealField(100)); 3*(Sqrt(25 + 10*Sqrt(5))); // G. C. Greubel, Nov 02 2018

evalf(3*(sqrt(25+10*sqrt(5))),130); # Muniru A Asiru, Nov 02 2018

RealDigits[3*Sqrt[25+10*Sqrt[5]],10,120][[1]] (* Harvey P. Dale, Jun 21 2011 *)

default(realprecision, 100); 3*(sqrt(25 + 10*sqrt(5))) \\ G. C. Greubel, Nov 02 2018

From Stanislav Sykora, Nov 30 2013: (Start)
Equals 15/tan(Pi/5).
Equals 15*phi/xi, where phi is the golden ratio (A001622) and xi its associate (A182007). (End)

More terms from Harvey P. Dale, Jun 21 2011

A179553 Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

			3.885540910050063539668319904270950058085880737273174114276853431338785...

Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 8

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552.

RealDigits[N[Sqrt[5/2*(10+Sqrt[5]+Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(10+sqrt(5)+sqrt(75+30sqrt(5))))/2.

A019889 Decimal expansion of sine of 80 degrees = cos(Pi/18).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.9848077530122080593667430245895230136706432517198424187900...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 6

Cf. A019849, A019859.

Digits:=100: evalf(sin(4*Pi/9)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[ Cos[Pi/18], 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[80 Degree],10,120][[1]] (* Harvey P. Dale, Apr 16 2022 *)

real(I^(1/9)) \\ Michel Marcus, Nov 29 2013

Equals 2F1(7/12,5/12;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Also the real part of I^(1/9). - Stanislav Sykora, Nov 29 2013
Equals sin(4*Pi/9). - Wesley Ivan Hurt, Sep 01 2014
Equals 2*A019849*A019859. - R. J. Mathar, Jan 17 2021
Largest positive root of 64*x^6 - 96*x^4 + 36*x^2 - 3. - Artur Jasinski, May 09 2025
Other roots are +- A019849 and +- A019829. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A010527. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/12,1/12; 1/2 ; 3/4). - R. J. Mathar, Aug 31 2025

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Dodecahedron: A three-dimensional figure with 12 faces, 20 vertices, and 30 edges.

Appears as a coordinate in a degree-7 quadrature formula on 12 points over the unit circle [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011

			1.40125853844407354467667793532206799444393173977549286366084518639135...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Chai Wah Wu, Table of n, a(n) for n = 1..10001
A. H. Stroud and Don Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Wikipedia, Dodecahedron.
Wolfram Alpha, Dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A102208, A102769, A131595, A179290, A179292, A179294.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

RealDigits[(Sqrt[3]+Sqrt[15])/4, 10, 175][[1]]

(1+sqrt(5))*sqrt(3)/4 \\ Stefano Spezia, Jan 27 2025

Equals (sqrt(3) + sqrt(15))/4 = sqrt((9 + 3*sqrt(5))/8).
The minimal polynomial is 16*x^4 - 36*x^2 + 9. - Joerg Arndt, Feb 05 2014
Equals (sqrt(3)/2) * phi = A010527 * A001622. - Amiram Eldar, Jun 02 2023

A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			16.5797497529881970460940463443632246181026360961176551818747440...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 8.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590.

RealDigits[N[(20+Sqrt[10*(80+31*Sqrt[5]+Sqrt[2175+930*Sqrt[5]])])/4,200]]

Digits of (20+sqrt(10*(80+31*sqrt(5)+sqrt(2175+930*sqrt(5)))))/4.

A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

Original entry on oeis.org

1, 4, 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

Offset: 2

Paolo Xausa, Jun 24 2025

The gyroelongated triangular bicupola is Johnson solid J_44.

			14.660254037844386467637231707529361834714026269...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated triangular bicupola.

Cf. A385256 (volume).
Cf. A002194, A010527.
Essentially the same of A332133, A375193 and A010527.

First[RealDigits[6 + 5*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J44", "SurfaceArea"], 10, 100]]

Equals 6 + 5*sqrt(3) = 6 + 5*A002194 = 6 + 10*A010527.

Equals the largest root of x^2 - 12*x - 39.

A020797 Decimal expansion of 1/sqrt(40).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With offset 1, decimal expansion of sqrt(5/2). - Eric Desbiaux, May 01 2008
sqrt(5/2) appears as a coordinate in a degree-5 integration formula on 13 points in the unit sphere [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
With offset 2, decimal expansion of sqrt(250). - Michel Marcus, Nov 04 2013
From Wolfdieter Lang, Nov 21 2017: (Start)
The regular continued fraction of 1/sqrt(40) = 1/(2*sqrt(10)) is [0; 6, 3, repeat(12, 3)], and the convergents are given by A(n-1)/B(n-1), n >= 0, with A(-1) = 0, A(n-1) = A041067(n) and B(-1) = 1, B(n-1) = A041066(n).
The regular continued fraction of sqrt(5/2) = sqrt(10)/2 is [1; repeat(1, 1, 2)], and the convergents are given in A295333/A295334.
sqrt(10)/2 is one of the catheti of the rectangular triangle with hypotenuse sqrt(13)/2 = A295330 and the other cathetus sqrt(3)/2 = A010527. This can be constructed from a regular hexagon inscribed in a circle with a radius of 1 unit. If the vertex V_0 has coordinates (x, y) = (1, 0) and the midpoint M_4 = (0, -sqrt(3)/2) then the point L = (sqrt(10)/2, 0) is obtained as intersection of the x-axis and a circle around M_4 with radius taken from the distance between M_4 and V_1 = (1/2, sqrt(3)/2) which is sqrt(13)/2. (End)

			1/sqrt(40) = 0.15811388300841896659994467722163592668597775696626084134287...
sqrt(5/2) = 1.5811388300841896659994467722163592668597775696626084134287...
sqrt(250) = 15.811388300841896659994467722163592668597775696626084134287...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. H. Stroud, D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2.

Cf. A010467 (sqrt(10)), A010527, A010494 (sqrt(40)), A041067/A041066, A295330, A295333/A295334.

RealDigits[N[1/Sqrt[40],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

1/sqrt(40) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(40*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals Re(sqrt(5*i)/10) = Im(sqrt(5*i)/10). - Karl V. Keller, Jr., Sep 01 2020

Equals A010467/20. - R. J. Mathar, Feb 23 2021

A179588 Decimal expansion of the surface area of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			11.56047793231506739113082378992526852408214900456427677440...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587.

Mathematica
```
RealDigits[N[7+2*Sqrt[2]+Sqrt[3],200]]
```

Digits of 7 + 2*sqrt(2) + sqrt(3).

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A232809 Decimal expansion of the surface index of a regular icosahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, surface area of a regular icosahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: A232812 (tetrahedron), 6.0 (cube = hexahedron), A232811 (octahedron), A232810 (dodecahedron), and this one.

An algebraic integer of degree 12 with minimal polynomial x^12 - 41115600x^6 + 765275040000. - Charles R Greathouse IV, Apr 25 2016

			5.14834855619951564633081294611601906410086411638672414845...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 12

Cf. A010527, A102208 (solid index of a sphere), A232808, A232810, A232811, A232812.

RealDigits[5*Sqrt[3]/(5*(3+Sqrt[5])/12)^(2/3), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

5*sqrt(3)/(5*(3+sqrt(5))/12)^(2/3) \\ Charles R Greathouse IV, Apr 25 2016

Equals 5*sqrt(3)/(5*(3+sqrt(5))/12)^(2/3).

Equals 10*A010527/A102208^(2/3).

cp's OEIS Frontend

A131595 Decimal expansion of 3*(sqrt(25 + 10*sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.

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A179553 Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

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A019889 Decimal expansion of sine of 80 degrees = cos(Pi/18).

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A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

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A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

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A020797 Decimal expansion of 1/sqrt(40).

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A131595 Decimal expansion of 3(sqrt(25 + 10sqrt(5))), the surface area of a regular dodecahedron with edges of unit length.