A381152 Decimal expansion of the isoperimetric quotient of a regular pentagon.

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

Offset: 0

Author

Paolo Xausa, Feb 15 2025

Keywords

• nonn
• cons
• easy

Comments

The isoperimetric quotient of a closed curve is equal to 4*Pi*A/p^2, where A is the area enclosed by the curve and p is its perimeter. For a regular n-gon, this is equivalent to Pi/(n*tan(Pi/n)).

The isoperimetric quotient of a circle is 1.

Examples

			0.86480626597720996723118206585862333703828555690228...
		

Links

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

Crossrefs

Cf. A019952, A102771.

Cf. isoperimetric quotient of other regular polygons: A073010 (triangle), A003881 (square), A093766 (hexagon), A381153 (heptagon), A196522 (octagon), A381154 (9-gon), A381155 (10-gon), A381156 (11-gon), A381157 (12-gon).

Programs

Mathematica

First[RealDigits[Pi/(5*Tan[Pi/5]), 10, 100]]

Formula

Equals Pi/(5*tan(Pi/5)) = (Pi/5)*A019952.

Equals (4/25)*Pi*A102771.