A200113 Decimal expansion of the area K of cyclic pentagon with sides 2, 3, 5, 7, and 11.

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

Offset: 2

Author

Zak Seidov, Nov 18 2011

Keywords

• nonn
• cons

Comments

s = {2,3,5,7,11}; R = circumradius (A200257).

K = sum((s(n)/4)(4*R^2-s(n)^2)^(1/2), n=1..5) = 41.9529923613808924252...

This is maximal area of pentagon with sides 2, 3, 5, 7, 11.

Crossrefs

Cf. A200257.

Programs

Mathematica

digits = 86; r = x /. FindRoot[ Sum[ ArcSin[ Prime[n]/2/x], {n, 5}] == Pi, {x, 5}, WorkingPrecision -> digits+10] // Re; Sum[p/4*(4*r^2 - p^2)^(1/2), {p, Prime /@ Range[5]}] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013, after Zak Seidov *)