A010527 -id:A010527 - cp's OEIS frontend

A179587 Decimal expansion of the volume of square cupola with edge length 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013
Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014
2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022

			1.942809041582063365867792482806465385713114583584632048784453158660...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Victor Oxman and Moshe Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geometricorum, Vol. 13 (2013), 113-115.
Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268.

Cf. A131594 (decimal expansion of sqrt(2)/3).

RealDigits[N[1+(2*Sqrt[2])/3,200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

sqrt(8)/3+1 \\ Charles R Greathouse IV, Nov 14 2016

Equals (3 + 2*sqrt(2))/3.

Equals 1 + 2*A131594. - L. Edson Jeffery, Nov 12 2014

A179292 Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

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

			0.794654472291766122955530928327594042026590588309264801754955775008438...

Mathforum, Icosahedron
Wikipedia, Icosahedron
MathWorld, Icosahedron
Index entries for algebraic numbers, degree 4.

Cf. A010527, A102208, A179290.

R=1;RealDigits[N[(Sqrt[75+30*Sqrt[5]]/15)*R,175]]

polrootsreal(45*x^4-30*x^2+1)[4] \\ Charles R Greathouse IV, Sep 02 2024

sqrt(6/sqrt(5)+3)/3 \\ Charles R Greathouse IV, Sep 02 2024

Sqrt(75 + 30*sqrt(5))/15.

Offset corrected, keyword:cons added by R. J. Mathar, Jul 11 2010

A179452 Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
Height of a regular pentagon with side length 1. - Jared Kish, Oct 16 2014
Volume of a regular decagonal prism with unit side length and height 2. - Wesley Ivan Hurt, May 04 2021

			1.53884176858762670128514528801845491200335107176889621351957812518743...

Eric Weisstein's World of Mathematics, Icosidodecahedron
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451.

Maple
```
sqrt(5+2*sqrt(5.))/2
```

RealDigits[Sqrt[5+2Sqrt[5]]/2,10,120][[1]] (* Harvey P. Dale, Jun 23 2017 *)

PARI
```
sqrt(5+2*sqrt(5))/2
```

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

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

Edited by M. F. Hasler, Oct 16 2014

A179552 Decimal expansion of the volume of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

Also equals the covariance-matrix eigenvalue of the regular icosahedron with unit edge lengths. - Chittaranjan Pardeshi, Jul 18 2025

			0.3015028323958245706837155695304698431433590983171469051779540518921...

Wikipedia, Covariance matrix.
Wikipedia, Regular icosahedron.
Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 2

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

Mathematica
```
RealDigits[N[(5+Sqrt[5])/24,200]]
```

(5+sqrt(5))/24 \\ Charles R Greathouse IV, Oct 30 2023

Equals (5+sqrt(5))/24.

A179449 Decimal expansion of the volume of great icosahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Great icosahedron: 20 faces, 12 vertices, and 30 edges.

			0.31830500937508762649617763802863490189974235016186428155379281441228294...

Eric Weisstein's World of Mathematics, Great Icosahedron
Wikipedia, Great Icosahedron
Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294.

RealDigits[N[5*(Sqrt[5]-3)/12, 105]][[1]]

Digits of 5/12 * (3-sqrt(5)).

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

A179451 Decimal expansion of the surface area of an icosidodecahedron with side length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			29.3059828449119895407453754361926770276025163091742830907641713815460...

Index entries for algebraic numbers, degree 8

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450.

RealDigits[N[Sqrt[30*(10+3*Sqrt[5]+Sqrt[75+30*Sqrt[5]])],200]]

polrootsreal(x^8 - 1200*x^6 + 324000*x^4 - 27000000*x^2 + 324000000)[8] \\ Charles R Greathouse IV, Oct 30 2023

Sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5))))

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

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

A179450 Decimal expansion of the volume of an icosidodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			13.83552593624940413982599206140528266708175201889932288543420886199647...

Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294, A179449.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/6,200]]
```

(45 + 17*sqrt(5))/6 \\ Charles R Greathouse IV, Oct 30 2023

(45 + 17*sqrt(5))/6.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A179590 Decimal expansion of the volume of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			2.32404531833319313093944911248751749029374557307435048284726483027368...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 2

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

Mathematica
```
RealDigits[N[(5+4*Sqrt[5])/6,200]]
```

Digits of (5+4*sqrt(5))/6.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A179587 Decimal expansion of the volume of square cupola with edge length 1.

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A179292 Decimal expansion of radius of inscribed sphere of an icosahedron with radius of circumscribed sphere = 1.

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A179452 Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

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Maple

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A179552 Decimal expansion of the volume of pentagonal pyramid with edge length 1.

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A179449 Decimal expansion of the volume of great icosahedron with edge length 1.

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Mathematica

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A179451 Decimal expansion of the surface area of an icosidodecahedron with side length 1.

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Mathematica

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Formula

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A010502 Decimal expansion of square root of 48.

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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.