A178817 -id:A178817 - cp's OEIS frontend

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Mar 30 2005

Equivalently, the area in the complex plane of the smallest convex set containing all order-6 roots of unity.
Subtracting 2.5 (i.e., dropping the first two digits) we obtain 0.09807.... which is a limiting mean cluster density for a bond percolation model at probability 1/2 [Finch]. - R. J. Mathar, Jul 26 2007
This constant is also the minimum radius of curvature of the exponential curve (occurring at x = -log(2)/2 = -0.34657359...). - Jean-François Alcover, Dec 19 2016
Luminet proves that this is the critical impact parameter of a bare black hole, in multiples of the Schwarzschild radius. That is, light from a distant source coming toward a black hole is captured by the black hole at smaller distances and deflected at larger distances. - Charles R Greathouse IV, May 21 2022
For any triangle ABC, sin(A) + sin(B) + sin(C) <= 3*sqrt(3)/2, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 16 2022
Surface area of a triangular bipyramid (Johnson solid J_12) with unit edges. - Paolo Xausa, Aug 04 2025

			2.59807621135331594029116951225880855041420788071557094208371046917789952536320...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

G. C. Greubel, Table of n, a(n) for n = 1..10000
S. R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat. vol 3 (1999) issue (2-4) pp. 323-335.
Kiran S. Kedlaya, A < B, (1999), Problem 6.1, p. 6.
J.-P. Luminet, Image of a spherical black hole with thin accretion disk, Astronomy and Astrophysics, vol. 75, no. 1-2 (May 1979), pp. 228-235.
Michael Penn, not as bad as it seems..., YouTube video, 2021.
Eric Weisstein et al., Root of Unity.
Eric Weisstein's World of Mathematics, de Moivre Number.
Eric Weisstein's World of Mathematics, Twenty-Vertex Entropy Constant.
Wikipedia, Hexagon.
Wikipedia, Regular polygon.
Wikipedia, Triangular bipyramid.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A104954, A104955, A104957.

Cf. Areas of other regular polygons: A120011, A102771, A178817, A090488, A256853, A178816, A256854, A178809.

Floor[n/2]*Sin[(2*Pi)/n] - Sin[(4*Pi*Floor[n/2])/n]/2 /. n -> 6
RealDigits[(3*Sqrt[3])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

3*sqrt(3)/2 \\ G. C. Greubel, Jul 03 2017

Equals (3*sqrt(3))/2, that is, 2*A104954.

Equals Product_{k>=3} (((k - 1)^2*(k + 2))/((k + 1)^2*(k - 2)))^(k/2). - Antonio Graciá Llorente, Oct 13 2024

A120011 Decimal expansion of sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Eric Desbiaux, Jul 04 2008

Area of equilateral triangle of side 1.
Quadratic number with denominator 4 and minimal polynomial 16x^2 - 3. - Charles R Greathouse IV, Jun 30 2021
With offset 1, surface area of a pentagonal bipyramid (Johnson solid J_13) with unit edges. - Paolo Xausa, Aug 04 2025

			0.43301270189221932338186158537646809173570131345259515701395....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Pentagonal bipyramid.
Index entries for algebraic numbers, degree 2

Cf. A010527.

Cf. Areas of higher regular polygons: A102771, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[Sqrt[3]/4,10,120][[1]] (* Harvey P. Dale, Feb 18 2016 *)

sqrt(3)/4 \\ Charles R Greathouse IV, Jun 26 2013

A090488 Decimal expansion of 2 + 2*sqrt(2).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A102771 Decimal expansion of area of a regular pentagon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Bryan Jacobs (bryanjj(AT)gmail.com), Feb 10 2005

			1.720477400588966922759011977...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. Areas of other regular polygons: A120011, A104956, A178817, A090488, A256853, A178816, A256854, A178809.

RealDigits[(5/4)*Sqrt[GoldenRatio^3/Sqrt[5]], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

5/(4*tan(Pi/5)) \\ Michel Marcus, Mar 25 2015

Equals sqrt(25 + 10*sqrt(5)) / 4.
Equals (3*phi+1)*sqrt(3-phi) with the golden section phi = (1 + sqrt(5))/2. - Wolfdieter Lang, Jan 25 2013
Equals 5/(4*tan(Pi/5)). - Michel Marcus, Mar 25 2015
Equals (5/4)*sqrt(phi^3/sqrt(5)). - G. C. Greubel, Jul 03 2017

Corrected the title. - Stanislav Sykora, Apr 12 2015

A256853 Decimal expansion of the area of a unit 9-gon.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 12 2015

From Michal Paulovic, May 09 2024: (Start)
This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.
9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

			6.181824193772900127213744059619763614941713348134358098386864...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Nonagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024

RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 9.
From Michal Paulovic, May 09 2024: (Start)
Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.
Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.
Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)
Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025

A178816 Decimal expansion of the area of the regular 10-gon (decagon) of edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

An algebraic number with degree 4 and denominator 2; minimal polynomial 16x^4 - 1000x^2 + 3125. - Charles R Greathouse IV, Apr 25 2016

This equals in a regular pentagon inscribed in a unit circle with vertices V0 = (x, y) = (1, 0), and V1..V4 in the counterclockwise sense, one tenth of the y-coordinate of the midpoint of side (V1,V2), named M1: M1_y = (2*sqrt(3 - phi) + sqrt(7 - 4*phi))/4 = sqrt(3 + 4*phi)/4. The x-coordinate is M1_x = -1/4. - Wolfdieter Lang, Jan 09 2018

			7.69420884293813350642572644009227456001675535884448106759789062593715...
sqrt(3 + 4*phi)/4 = 0.769420884293813350642572644009227456001675535884... - _Wolfdieter Lang_, Jan 09 2018

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Wikipedia, Decagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 4

Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A090488, A256853, A256854, A178809.

Cf. A001622.

SetDefaultRealField(RealField(100)); 5*Sqrt(2*Sqrt(5)+5)/2; // G. C. Greubel, Jan 22 2019

evalf[120](5*sqrt(5+2*sqrt(5))/2); # Muniru A Asiru, Jan 22 2019

RealDigits[5*Sqrt[5+2*Sqrt[5]]/2, 10, 100][[1]]

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

numerical_approx(5*sqrt(2*sqrt(5)+5)/2, digits=100) # G. C. Greubel, Jan 22 2019

Digits of 5*sqrt(5+2*sqrt(5))/2 = (5/2)*sqrt(3 + 4*phi), with phi from A001622.

A256854 Decimal expansion of area of a regular 11-gon with unit edge length.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 12 2015

			9.36563990694543752488235845328433428788257496183502738768931...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Hendecagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 10

Cf. A000796, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A256853 (p=9), A178816 (p=10), A178809 (p=12).

RealDigits[11/4 Cot[Pi/11],10,120][[1]] (* Harvey P. Dale, Apr 03 2016 *)

p=11; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 11.

A374972 Decimal expansion of the sagitta of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 26 2024

			0.114121737195074969038805681030507391369390840490...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta.
Index entries for algebraic numbers, degree 6.

Cf. A374957 (circumradius), A374971 (apothem), A178817 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375068 (pentagon), A375069 (hexagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A343059.

First[RealDigits[Tan[Pi/14]/2, 10, 100]]

tan(Pi/14)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(448*x^6-560*x^4+84*x^2-1)[4] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/14)/2 = A343059/2.

Equals A374957 - A374971.

A374971 Decimal expansion of the apothem (inradius) of a regular heptagon with unit side length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 26 2024

			1.0382606982861682835817694307429201653528601033...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 6.

Cf. A374957 (circumradius), A374972 (sagitta), A178817 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A375067 (pentagon), A010527 (hexagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A178818, A073052, A343058.

Mathematica
```
First[RealDigits[Cot[Pi/7]/2, 10, 100]]
```

 5/(Pi/7) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/7)/2 = A178818/2.
Equals 1/(2*tan(Pi/7)) = 1/(2*A343058).
Equals A374957*cos(Pi/7) = A374957*A073052.
Equals A374957 - A374972.
Largest of the 6 real-valued roots of 448*x^6 -560*x^4 +84*x^2 -1 =0. - R. J. Mathar, Aug 29 2025

A381153 Decimal expansion of the isoperimetric quotient of a regular heptagon.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Feb 15 2025

For the definition of isoperimetric quotient, see A381152.

			0.93194062349909574595222630089422754574528525154715...

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

Cf. A178817, A343058.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A381152 (pentagon), A093766 (hexagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

First[RealDigits[Pi/(7*Tan[Pi/7]), 10, 100]]

Equals Pi/(7*tan(Pi/7)) = Pi/(7*A343058).

Equals (4/49)*Pi*A178817.

cp's OEIS Frontend

A104956 Decimal expansion of the area of the regular hexagon with circumradius 1.

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A120011 Decimal expansion of sqrt(3)/4.

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A090488 Decimal expansion of 2 + 2*sqrt(2).

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A102771 Decimal expansion of area of a regular pentagon with unit edge length.

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A256853 Decimal expansion of the area of a unit 9-gon.

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A178816 Decimal expansion of the area of the regular 10-gon (decagon) of edge length 1.

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A256854 Decimal expansion of area of a regular 11-gon with unit edge length.

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