A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

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

Offset: 1

Author

Amiram Eldar, Nov 12 2021

Keywords

• nonn
• cons

Comments

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

Examples

			1.12256994144896343110486287949381696894803120580270...
		

References

• Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Links

• Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
• Mathematics Stack Exchange, Area of a five pointed star, 2014.
• Wikipedia, Pentagram.
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

Programs

Mathematica

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Formula

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).

Equals 5/(cot(Pi/5) + cot(Pi/10)).

Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).

Equals (5/2)*sqrt((25 -11*sqrt(5))/2).

Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.