A002194 -id:A002194 - cp's OEIS frontend

A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

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

Offset: 0

Stanislav Sykora, May 02 2016

Since 20-gon is constructible (see A003401), this is a constructible number.

			0.3128689300804617380202106389343337846277997841713215801692826921...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A003401.
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17).
Cf. A019818.

RealDigits[N[2Sin[Pi/20], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/20)
```

Equals 2*sin(Pi/20) = 2*A019818.
Equals also (sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5)))/4.
Equals i^(9/10) + i^(-9/10). - Gary W. Adamson, Jul 08 2022

A293328 Least integer k such that k/2^n > sqrt(1/3).

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 37, 74, 148, 296, 592, 1183, 2365, 4730, 9460, 18919, 37838, 75675, 151349, 302698, 605396, 1210792, 2421583, 4843166, 9686331, 19372661, 38745321, 77490642, 154981283, 309962566, 619925132, 1239850263, 2479700525, 4959401050, 9918802099

Offset: 0

Clark Kimberling, Oct 10 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002194, A094386, A293327, A293329.

z = 120; r = Sqrt[1/3];
Table[Floor[r*2^n], {n, 0, z}];   (* A293327 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293328 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293329 *)

a(n) = ceiling(r*2^n), where r = sqrt(1/3).

a(n) = A293327(n) + 1.

A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 16 2022

The solution to a problem in Jones (1932): "At each corner of a garden, surrounded by a wall n yards square, a goat is tied with a rope n yards long. Find the area of the part of the garden common to the four goats." (When the square is taken to be of unit size, the common area is this constant.)
The perimeter of the shape formed by the intersection is 2*Pi/3 (A019693).
The solution to the three-dimensional version of this problem is A352454.

			0.31514674362772045262676811958729526112291787931465...

Donald L. Chambers, Problem 3684, School Science and Mathematics, Vol. 77, No. 5 (1977), p. 443; Solution by J. Philip Smith, ibid., Vol. 78, No. 4 (1978), pp. 354-355.
Amiram Eldar, Illustration.
Samuel Isaac Jones, Mathematical Nuts: For Lovers of Mathematics, 1932, Problems 9 and 10, pp. 86, 301-302.
Missouri State University, Problem #8, Finding the Area (resp. Volume) of Overlapping Circles (resp. Spheres), Advanced Problem Archive,; Solution to Problem #8, by Raymond Roan.
Bruce Shawyer, Problem 6, APICS 1999 Mathematics Competition, The Academy Corner, Crux Mathematicorum, Vol. 25, No. 8, 1999, p. 453; Solutions by Richard Tod and Catherine Shevlin, Vol. 26, No. 4, 2000, pp. 193-194.
Charles W. Trigg, Problem 686, Crux Mathematicorum, Vol. 7, No. 9, 1981, p. 275; Solution by Jordan Dou, Vol. 8, No. 9, 1982, p. 294.

Cf. A002194, A019670, A019693.

Cf. A075838, A133731, A192930, A352454.

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

Equals 1 + Pi/3 - sqrt(3) = 1 + A019670 - A002194.

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 06 2024

			72.60725303413392187893153397383948620117264765443...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377751 (volume), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A010527 (analogous for a regular icosahedron, with offset 1).
Cf. A002163, A002194, A375067.

First[RealDigits[3*(10*Sqrt[3] + Sqrt[25 + Sqrt[500]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "SurfaceArea"], 10, 100]]

3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) \\ Charles R Greathouse IV, Feb 05 2025

Equals 3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) = 30*A002194 + 3*sqrt(25 + 10*A002163).

Equals 30*(A002194 + A375067).

A384872 Decimal expansion of the surface area of a pentagonal orthocupolarotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 11 2025

The pentagonal orthocupolarotunda is Johnson solid J_32.

Also the surface area of a pentagonal gyrocupolarotunda (Johnson solid J_33) with unit edge.

			23.538532332506058310041007622367288571887138891860...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Pentagonal gyrocupolarotunda.
Wikipedia, Pentagonal orthocupolarotunda.

Cf. A384871 (volume).

Cf. A002163, A002194, A384284, A384286, A384625.

First[RealDigits[5 + 15/4*Sqrt[3] + 7/4*Sqrt[25 + 10*Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J32", "SurfaceArea"], 10, 100]]

Equals 5 + (15/4)*sqrt(3) + (7/4)*sqrt(25 + 10*sqrt(5)) = 5 + (15/4)*A002194 + (7/4)*sqrt(25 + 10*A002163).

Equals the largest root of 256*x^8 - 10240*x^7 + 57600*x^6 + 1856000*x^5 - 21756000*x^4 + 6320000*x^3 + 484812500*x^2 - 364125000*x - 342171875.

A385261 Decimal expansion of the surface area of a gyroelongated pentagonal bicupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 27 2025

The gyroelongated pentagonal bicupola is Johnson solid J_46.

			26.431335857944513546973871516071261950885787743598...

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

Cf. A385260 (volume).

Cf. A002163, A002194, A384284, A385257, A385259.

First[RealDigits[(20 + 15*Sqrt[3] + Sqrt[25 + 10*Sqrt[5]])/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J46", "SurfaceArea"], 10, 100]]

Equals (20 + 15*sqrt(3) + sqrt(25 + 10*sqrt(5)))/2 = (20 + 15*A002194 + sqrt(25 + 10*A002163))/2.

Equals the largest root of x^8 - 80*x^7 + 2100*x^6 - 14000*x^5 - 174750*x^4 + 1390000*x^3 + 9603125*x^2 + 9937500*x - 6546875.

A385696 Decimal expansion of the surface area of an augmented dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 08 2025

The augmented dodecahedron is Johnson solid J_58.

			21.090314915939732767258439678157046052159622437375...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Augmented dodecahedron.

Cf. A385695 (volume).

Cf. A002194, A010476, A385803, A385805.

First[RealDigits[(5*Sqrt[3] + 11*Sqrt[25 + 10*Sqrt[5]])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J58", "SurfaceArea"], 10, 100]]

Equals (5*sqrt(3) + 11*sqrt(5*(5 + 2*sqrt(5))))/4 = (5*A002194 + 11*sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 198400*x^6 + 41204000*x^4 - 1620087500*x^2 + 7460640625.

A386465 Decimal expansion of the surface area of an augmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 25 2025

The augmented truncated dodecahedron is Johnson solid J_68.

			102.18209222021391857798854245281533205298421595361...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Augmented truncated dodecahedron.

Cf. A386464 (volume).

Cf. A002194, A010476, A377694, A386412, A386460, A386543, A386545.

First[RealDigits[(20 + 25*Sqrt[3] + 110*Sqrt[#] + Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J68", "SurfaceArea"], 10, 100]]

Equals (20 + 25*sqrt(3) + 110*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5))))/4 = (20 + 25*A002194 + 110*sqrt(5 + A010476) + sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 10240*x^7 - 3955200*x^6 + 122240000*x^5 + 16152924000*x^4 - 343551280000*x^3 - 11461251137500*x^2 + 131995515375000*x + 634637481578125.

A386543 Decimal expansion of the surface area of a parabiaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 28 2025

The parabiaugmented truncated dodecahedron is Johnson solid J_69.

Also the surface area of a metabiaugmented truncated dodecahedron (Johnson solid J_70) with unit edges.

			103.37342428732584861123113591699400755105334133204...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Metabiaugmented truncated dodecahedron.
Wikipedia, Parabiaugmented truncated dodecahedron.

Cf. A386466 (volume).

Cf. A002194, A010476, A377694, A385803, A386465, A386545.

First[RealDigits[(20 + 15*Sqrt[3] + 50*Sqrt[#] + Sqrt[5*#])/2 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J69", "SurfaceArea"], 10, 100]]

Equals (20 + 15*sqrt(3) + 50*sqrt(5 + 2*sqrt(5)) + sqrt(5*(5 + 2*sqrt(5))))/2 = (20 + 15*A002194 + 50*sqrt(5 + A010476) + sqrt(5*(5 + A010476)))/2.

Equals the largest root of x^8 - 80*x^7 - 11400*x^6 + 796000*x^5 + 31475250*x^4 - 1804610000*x^3 - 8296459375*x^2 + 548931187500*x - 2544044046875.

A386545 Decimal expansion of the surface area of a triaugmented truncated dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Jul 28 2025

The triaugmented truncated dodecahedron is Johnson solid J_71.

			104.56475635443777864447372938117268304912246671047...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Wikipedia, Triaugmented truncated dodecahedron.

Cf. A386544 (volume).

Cf. A002194, A010476, A377694, A385805, A386465, A386543.

First[RealDigits[(60 + 35*Sqrt[3] + 90*Sqrt[#] + 3*Sqrt[5*#])/4 & [5 + Sqrt[20]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J71", "SurfaceArea"], 10, 100]]

Equals (60 + 35*sqrt(3) + 90*sqrt(5 + 2*sqrt(5)) + 3*sqrt(5*(5 + 2*sqrt(5))))/4 = (60 + 35*A002194 + 90*sqrt(5 + A010476) + 3*sqrt(5*(5 + A010476)))/4.

Equals the largest root of 256*x^8 - 30720*x^7 - 1574400*x^6 + 238464000*x^5 + 68364000*x^4 - 390828240000*x^3 + 4437895162500*x^2 + 78660973125000*x - 1021409416546875.

cp's OEIS Frontend

A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

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A293328 Least integer k such that k/2^n > sqrt(1/3).

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A352453 Decimal expansion of the area of intersection of 4 unit-radius circles that have the vertices of a unit-side square as centers.

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A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

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A384872 Decimal expansion of the surface area of a pentagonal orthocupolarotunda with unit edge.

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A385261 Decimal expansion of the surface area of a gyroelongated pentagonal bicupola with unit edge.

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A385696 Decimal expansion of the surface area of an augmented dodecahedron with unit edge.

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A386465 Decimal expansion of the surface area of an augmented truncated dodecahedron with unit edges.

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