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

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

Author

Vladimir Joseph Stephan Orlovsky, Jun 16 2010

Keywords

• nonn
• cons
• easy

Comments

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

Examples

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

Links

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

Crossrefs

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

Cf. A001622.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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