A019699 -id:A019699 - cp's OEIS frontend

A164104 Decimal expansion of 8*Pi^2/3.

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 5-dimensional unit sphere.

			Equals 26.318945069571622983558642666336...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A019692, A019694, A164102, A091925, A164107, A164109.

RealDigits[8*Pi^2/3, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

PARI
```
8*Pi^2/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 5*A164103 = 10 * A019699 * A019692.

A-number in formula corrected by R. J. Mathar, Aug 12 2010

A164105 Decimal expansion of Pi^3/6.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 6-dimensional unit sphere.

			Equals 5.1677127800499700292460525111835658670375480943...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164106, A164108.

RealDigits[Pi^3/6, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^3/6 \\ G. C. Greubel, Apr 09 2017
```

Equals A091925/6 = A019670*A102753.

A258146 Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, May 31 2015

The formula for the ratio of the area of a circular segment with central angle alpha and the area of one half of the corresponding circular disk is (alpha - sin(alpha))/Pi. Here alpha = Pi/2.
This is also the ratio of the area of a circular disk without a central inscribed rectangle (2*x, 2*y) together with the two opposite circular segments each with central angle beta and the area of the circular disk. This is the analog of the ratio of the volume of a sphere with missing central cylinder symmetric hole of length 2*y and the area of the sphere. See a comment on A019699. In two dimensions this problem is not remarkable, because the radius R of the circle does matter. The formula is here: area ratio ar = 1 - (beta + sin(beta)/Pi) where beta = arcsin(2*yhat*sqrt(1-yhat^2)), with yhat = y/R, and beta = Pi - alpha from above.
The astonishing result from three dimensions, ar_3 = yhat^3, could suggest ar = yhat^2, which is wrong. Thanks to Sven Heinemeyer for inspiring me to look into this.
Essentially the same digit sequence as A188340. - R. J. Mathar, Jun 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Circular Segment.
Index entries for transcendental numbers

Cf. A019699, A060294, A258147, A258148.

RealDigits[(1-2/Pi)/2,10,120][[1]] (* Harvey P. Dale, Sep 23 2017 *)

1/2 - 1/Pi \\ Michel Marcus, Oct 19 2017

Area ratio ar = (1 - 2/Pi)/2 = 0.181690113816209...

For Buffon's constant 2/Pi see A060294.

A276023 Decimal expansion of 32*Pi^4/945.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 16 2016

Volume of the 9-dimensional unit sphere.

More generally, the ordinary generation function for the volume of the n-dimensional unit sphere is exp(Pi*x^2)*(erf(sqrt(Pi)*x) + 1) = 1 + 2*x + Pi*x^2 + (4*Pi/3)*x^3 + (Pi^2/2)*x^4 + ...

			3.2985089027387068693821065037445117...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Volume of the n-dimensional unit sphere
Eric Weisstein's World of Mathematics, Hypersphere
Index entries for transcendental numbers

Cf. A021949, A092425.

Cf. similar sequences of the volume of the n-dimensional unit sphere: A000796 (n = 2), 10*A019699 (n = 3), A102753 (n = 4), A164103 (n = 5), A164105 (n = 6), A164106 (n = 7), A164108 (n = 8).

Mathematica
```
RealDigits[(32 Pi^4)/945, 10, 120][[1]]
```

(32*Pi^4)/945 \\ G. C. Greubel, Apr 09 2017

Equals 32*A092425*A021949.

A374771 Decimal expansion of the volume of the sphere inscribed in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 19 2024

			5.78333595039657417842182321041033675553722324626...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Regular dodecahedron.

Cf. A000796, A001622, A019699, A102769, A237603 (radius), A374772.

First[RealDigits[Pi*Sqrt[1525 + 682*Sqrt[5]]/30, 10, 100]]

Equals (4/3)*Pi*A237603^3 = 10*A019699*A237603^3.
Equals (1/30)*Pi*sqrt(1525 + 682*sqrt(5)).
Equals (Pi/6)*A001622^6/((3 - A001622)^(3/2)).
Equals A102769*A374772.

A336198 Decimal expansion of the radius of a sphere centered on the surface of a unit-radius sphere and dividing it into two parts of equal volume.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 11 2020

The solution to the grazing goat problem in three dimensions.

			1.228544863735220903448994497685293465644191645518602...

Marshall Fraser, The Grazing Goat in n Dimensions, The Two-Year College Mathematics Journal, Vol. 15, No. 2 (1984), pp. 126-134.
Mark D. Meyerson, Return of the Grazing Goat in n Dimensions, The College Mathematics Journal, Vol. 15, No. 5 (1984), pp. 430-432.
Eric Weisstein's World of Mathematics, Goat Problem.
Wikipedia, Goat problem.

Cf. A019699, A133731.

RealDigits[x /. Solve[3*x^4 - 8*x^3 + 8 == 0 && x > 0, {x}, Reals][[1]], 10, 100][[1]]

The smaller of the 2 real roots of the equation 3*x^4 - 8*x^3 + 8 = 0.

A270230 Decimal expansion of 3/(4*Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Mar 13 2016

Consider generic prisms with triangular bases (tp), enclosed by a sphere, and let f(tp) be the fraction of the sphere volume occupied by any of them (i.e., the ratio of the prism volume to the sphere volume). Then this constant is the supremum of f(tp). It is attained by prisms which have as their base equilateral triangles with edge lengths r*sqrt(2), and rectangular side faces that are r*sqrt(2) wide and r*2/sqrt(3) high, where r is the radius of the enclosing, circumscribed sphere.
An intriguing fact is that the volume of such a best-fitting prism is exactly r^3. Hence, 1/a is the volume of a sphere with radius 1.
Examples of similar constants obtained for other shapes enclosed by spheres are: A020760 for cylinders and A165952 for cuboids.

			0.238732414637843003653325645058771543051689468610684673121501016...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Index entries for transcendental numbers

Cf. A002193, A019699 (one tenth of 1/a), A020760, A020832 (one tenth of 2/sqrt(3)), A165952.

First@ RealDigits[N[3/4/Pi, 120]] (* Michael De Vlieger, Mar 15 2016 *)

PARI
```
3/4/Pi
```

A273264 Volume of unit n-ball, rounded to the nearest integer.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 5, 4, 3, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Felix Fröhlich, May 18 2016

			The volume of the unit 3-ball (the set of points of distance equal to or less than the radius of the midpoint of the ordinary sphere or 2-sphere) is equal to 4/3*Pi, which is 4.1887902047... (A019699) and when rounded down this is 4, so a(3) = 4.

Wikipedia, Volume of an n-ball.

Cf. A019699, A177677, A228272.

Table[Round[(Pi^(n/2))/Gamma[1 + n/2]], {n, 120}] (* Michael De Vlieger, May 19 2016 *)

a(n) = round((Pi^(n/2)) / (gamma(1+n/2)))

a(n) = round((Pi^(n/2)) / (Gamma(1 + n/2))).

A210962 Decimal expansion of 4*(2 - Pi/3).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 03 2012

Volume between a sphere of radius 1 and the circumscribed cube.

			3.8112097952136090153831...

Index entries for transcendental numbers

Cf. A019670, A019699, A153799.

RealDigits[4 (2 - Pi/3), 10, 87][[1]] (* Bruno Berselli, Aug 03 2012 *)

8 - 4/3*Pi \\ Charles R Greathouse IV, Oct 01 2022

4*(2 - Pi/3) = 8 - 4*Pi/3 = 8 - A019699.

A321463 Decimal expansion of 36*Pi.

Original entry on oeis.org

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

Offset: 3

Felix Fröhlich, Nov 10 2018

Surface area and volume of a sphere of radius 3, the unique non-degenerate sphere with volume equal to surface area.
Let r be the radius of the sphere. Set (4/3)*Pi*r^3 = 4*Pi*r^2, then (4/3)*Pi*r = 4*Pi and r = 3. Thus, the volume V(3) = (4/3)*Pi*3^3 = 36*Pi and the surface area A(3) = 4*Pi*3^2 = 36*Pi.
In other words: 36*Pi is also the surface area of a sphere whose diameter equals the square root of 36. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Nov 10 2018

			113.097335529232556584655161798062103831098098377503809555098005323081390626....

Index entries for transcendental numbers

Cf. A000796.

Cf. A019694 (surface area of sphere of radius 1), A019699 (volume of sphere of radius 1).

First[RealDigits[N[36*Pi, 100], 10]] (* Stefano Spezia, Nov 10 2018 *)

PARI
```
36*Pi
```

Equals 36*A000796.

cp's OEIS Frontend

A164104 Decimal expansion of 8*Pi^2/3.

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A164105 Decimal expansion of Pi^3/6.

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A258146 Decimal expansion of (1 - 2/Pi)/2: ratio of the area of a circular segment with central angle Pi/2 and the area of the corresponding circular half-disk.

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A276023 Decimal expansion of 32*Pi^4/945.

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A374771 Decimal expansion of the volume of the sphere inscribed in a regular dodecahedron with unit edge.

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A336198 Decimal expansion of the radius of a sphere centered on the surface of a unit-radius sphere and dividing it into two parts of equal volume.

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A270230 Decimal expansion of 3/(4*Pi).

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