A102771 -id:A102771 - cp's OEIS frontend

A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

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

Offset: 0

Paolo Xausa, Feb 15 2025

The isoperimetric quotient of a closed curve is equal to 4*Pi*A/p^2, where A is the area enclosed by the curve and p is its perimeter. For a regular n-gon, this is equivalent to Pi/(n*tan(Pi/n)).

The isoperimetric quotient of a circle is 1.

			0.86480626597720996723118206585862333703828555690228...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Isoperimetric Quotient.
Wikipedia, Isoperimetric inequality.

Cf. A019952, A102771.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(5*Tan[Pi/5]), 10, 100]]

Equals Pi/(5*tan(Pi/5)) = (Pi/5)*A019952.

Equals (4/25)*Pi*A102771.

A255605 Integer part of the area of a pentagon with side length n.

Original entry on oeis.org

1, 6, 15, 27, 43, 61, 84, 110, 139, 172, 208, 247, 290, 337, 387, 440, 497, 557, 621, 688, 758, 832, 910, 990, 1075, 1163, 1254, 1348, 1446, 1548, 1653, 1761, 1873, 1988, 2107, 2229, 2355, 2484, 2616, 2752, 2892, 3034, 3181, 3330, 3483, 3640, 3800, 3963, 4130, 4301, 4474

Offset: 1

Kival Ngaokrajang, Feb 27 2015

nonn

Column 3 of A255604.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A102771, A255604.

Table[IntegerPart[5*n^2/(4*Tan[Pi/5])], {n, 51}] (* Michael De Vlieger, Mar 18 2015 *)
With[{c=Sqrt[25+10*Sqrt[5]]/4},Table[IntegerPart[c*n^2],{n,60}]] (* Harvey P. Dale, Aug 17 2020 *)

{for(n=1,100,a=floor(5*n^2/(4*tan(Pi/5)));print1(a,", "))}

my(c=25+10*quadgen(20)); a(n) = sqrtint(floor(c*n^4))>>2; \\ Kevin Ryde, May 07 2021

a(n) = floor(5*n^2/(4*tan(Pi/5))), n >= 1.

A386001 Decimal expansion of the surface area of a tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 17 2025

The tridiminished icosahedron is Johnson solid J_63.

			7.32649571122799738518634385904816925690062907729...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tridiminished icosahedron.

Cf. A386000 (volume).

Cf. A002194, A010476, A102771, A120011, A386003.

First[RealDigits[(5*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J63", "SurfaceArea"], 10, 100]]

Equals (5*sqrt(3) + 3*sqrt(5*(5 + 2*sqrt(5))))/4 = (5*A002194 + 3*sqrt(5*(5 + A010476)))/4.
Equals 3*A102771 + 5*A120011 = A386003 - 2*A120011.
Equals the largest root of 256*x^8 - 19200*x^6 + 324000*x^4 - 1687500*x^2 + 1265625.

A386003 Decimal expansion of the surface area of an augmented tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 18 2025

The augmented tridiminished icosahedron is Johnson solid J_64.

			8.192521115012436031950067029801105440372031704...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented tridiminished icosahedron.

Cf. A386002 (volume).

Cf. A002194, A010476, A102771, A120011, A386001.

First[RealDigits[(7*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J64", "SurfaceArea"], 10, 100]]

Equals (7*sqrt(3) + 3*sqrt(5*(5 + 2*sqrt(5))))/4 = (7*A002194 + 3*sqrt(5*(5 + A010476)))/4.
Equals 3*A102771 + 7*A120011 = A386001 + 2*A120011.
Equals the largest root of 256*x^8 - 23808*x^6 + 484704*x^4 - 2752812*x^2 + 4626801.

A179050 Decimal expansion of 5/(2sqrt(5+2sqrt(5))), area of regular pentagram with base edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 26 2010

An algebraic number of degree 4: the smaller positive root of 16x^4 - 200x^2 + 125. - Charles R Greathouse IV, Dec 03 2012

			0.81229924058226581538967853053783616238725867880368775076951179784168...

Thomas M. Green, Area of Pentagram
Index entries for algebraic numbers, degree 4

Cf. A002194, A010502, A102771, A120011, A179047, A179048, A104956.

a=1;area=5/(2*Sqrt[5+2*Sqrt[5]]);RealDigits[N[area,20]]

5/sqrt(20+8*sqrt(5)) \\ Charles R Greathouse IV, Dec 03 2012

Offset corrected, keyword:cons inserted by R. J. Mathar, Jun 28 2010

Name corrected by Charles R Greathouse IV, Dec 03 2012

A220674 Decimal expansion of the area of Dürer's approximation of a regular pentagon with each side of unit length.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jan 30 2013

To be read as dimensionless area F_D/r^2 = 1.720311429... where the length of each side is r, which is the radius of each circle in Dürer's construction. See the link Dürer, Zweites Buch, figure 16. Compare this with the regular pentagon with unit side length, which is given in A102771, and is F_5/r^2 = 1.720477400... The relative error is about -0.96*10^{-4}.
The angles in Dürer's pentagon are approximately twice 108.3661201 degrees, twice 107.0378260 degrees and once 109.1921079 degrees. The sum has to be exactly 3*Pi, or 540 degrees, as for any pentagon. For the analytic values see the W. Lang link.
Alonso del Arte pointed out the Hughes reference where this construction is shown on p. 5 and p. 16. See also the historical remarks on p. 17.
In the cut-the-knot link this construction is considered in more detail, and the two interior angles at the bottom of the pentagon are shown to be 108.36612... degrees.
For more references and links see the W. Lang link. The length of the dimensionless diagonals which approximate the golden section are also given there, and the angles of the companion of Dürer's pentagon with the same area are computed there. - Wolfdieter Lang, Feb 14 2013

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cut-The-Knot, Approximate Construction of Regular Pentagon by A. Dürer
G. Hughes, The Polygons of Albrecht Durer -1525, arXiv:1205.0080 [math.HO], 2012.
Wolfdieter Lang, Albrecht Dürer's approximation of a regular 5-gon.
Wikimedia Commons, Albrecht Dürer, Underweysung der messung ..., 1525, title page.
Wikisource Albrecht Dürer, Underweysung der messung ..., Zweites Buch, 1525 (in German).

Cf. A102771 (pentagon area).

r = Sqrt[7 - 3*Sqrt[3] + 2*(Sqrt[3]-1)* Cos[a]]; area = (2 + Sqrt[3] + r - 2*Cos[a]*(r+2))/4 /. Cos[a] -> (3 - Sqrt[3] - Sqrt[6*Sqrt[3]-4])/4; RealDigits[area, 10, 100] // First (* Jean-François Alcover, Feb 13 2013 *)

The dimensionless area of Dürer's pentagon is

F_D/r^2 = (1 + 2*x)*y1/2 + x*y2, with x = (a + sqrt(a*(a+8)))/4, a := sqrt(3) - 1, y1 = 1 - sqrt(3)/2 + x, y2 = sqrt(1 - x^2). The approximate values for x, y1 and y2 are 0.8150878978, 0.9490624938, 0.5793373101, respectively. This leads to the approximate value 1.720311430 for F_D/r^2, and the present sequence gives more accurate digits.

A384036 Decimal expansion of the surface area of a regular pentagonal prism of edge length 1.

Original entry on oeis.org

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

Offset: 1

Kritsada Moomuang, May 17 2025

			8.4409548011779338455...

Cf. A178809.

Cf. A102771 (volume), A300074 (midradius), A384059 (circumradius).

RealDigits[5 + 1/2 * Sqrt(25 + 10 * Sqrt(5)), 10, 100, 0][[1]]

Equals 5 + (1/2)*sqrt(25 + 10*sqrt(5)).

Minimal polynomial: 16*x^4 - 320*x^3 + 2200*x^2 - 6000*x + 5125. - Stefano Spezia, May 17 2025

A384059 Decimal expansion of the circumradius of a regular pentagonal prism of edge length 1.

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, May 18 2025

			0.98671515532598310...

Cf. A102771 (volume), A300074 (midradius), A384036 (surface area).

RealDigits[1/2 * Sqrt(3 + 2/Sqrt(5)), 10, 100, -1][[1]]

Equals (1/2)*sqrt(3 + 2/sqrt(5)).

Minimal polynomial: 80*x^4 - 120*x^2 + 41. - Stefano Spezia, May 18 2025

A323098 Decimal expansion of edge length of a regular pentagon with area 1.

Original entry on oeis.org

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

Offset: 0

Jinyuan Wang, Oct 26 2019

			0.7623870555067738744984026821...

Cf. A102771, A154605 (the edge length of an equilateral triangle).

RealDigits[(2)/(25 + 10*Sqrt[5])^(1/4), 10, 100][[1]] (* Metin Sariyar, Oct 26 2019 *)

default(realprecision, 110); 2/(25 + 10*sqrt(5))^(1/4)

Equals sqrt(1/A102771) = 2/(25 + 10*sqrt(5))^(1/4).

cp's OEIS Frontend

A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

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A255605 Integer part of the area of a pentagon with side length n.

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A386001 Decimal expansion of the surface area of a tridiminished icosahedron with unit edge.

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A386003 Decimal expansion of the surface area of an augmented tridiminished icosahedron with unit edge.

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A179050 Decimal expansion of 5/(2*sqrt(5+2*sqrt(5))), area of regular pentagram with base edge length 1.

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A220674 Decimal expansion of the area of Dürer's approximation of a regular pentagon with each side of unit length.

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A384036 Decimal expansion of the surface area of a regular pentagonal prism of edge length 1.

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A384059 Decimal expansion of the circumradius of a regular pentagonal prism of edge length 1.

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A179050 Decimal expansion of 5/(2sqrt(5+2sqrt(5))), area of regular pentagram with base edge length 1.