A394451 - OEIS

%I #13 Apr 24 2026 13:37:23

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

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

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

%N Decimal expansion of the area of a 20-gon with unit side lengths.

%C The largest solution to x^4 - 20*x^3 - 350*x^2 - 500*x + 625 = 0.

%C This constant multiplied by the square of the side length of a regular 20-gon equals the area of that 20-gon.

%C 20^2 divided by this constant equals 80 * tan(Pi/20) = 12.67075522... which is the perimeter and the area of an equable 20-gon with its side length 4 * tan(Pi/20) = 0.63353776... .

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Icosagon">Icosagon</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.

%F Equals 5 / tan(Pi/20).

%F Equals 5 / (1 + sqrt(5) - sqrt(5 + sqrt(20))).

%F Equals 5 * tan(9*Pi/20).

%F Equals 5 * (1 + sqrt(5) + sqrt(5 + sqrt(20))).

%F Equals 5 + 10 * (sqrt(5/4) + sqrt(5/4 + sqrt(5/4))) = 5 + 10 * A188594.

%F Equals 5 * phi * (2 + sqrt(5/2 + sqrt(5/4))) where phi = 1.61803398... .

%e 31.5687575733752154...

%p evalf((5 / tan(Pi/20)), 105);

%t RealDigits[5 / Tan[Pi/20], 10, 105][[1]]

%o (PARI) 5 / tan(Pi/20)

%Y Cf. A001622, A019907, A019979, A188594.

%K nonn,cons

%O 2,1

%A _Michal Paulovic_, Mar 20 2026