A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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

Offset: 0

Author

Michal Paulovic, Dec 08 2022

Keywords

• nonn
• cons

Comments

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:

A = (R^2 / 2) * (theta - sin(theta))

A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))

A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)

A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...

where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.

The sagitta (height) of the circular segment is:

h = R * (1 - cos(theta/2))

h = 1 * (1 - cos(Pi/3))

h = 1 - 1/2 = 0.5 (A020761)

Examples

			0.6141848493043784...
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Programs

Maple

evalf(Pi/3-sqrt(3)/4);

Mathematica

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI

Pi/3 - sqrt(3)/4

Formula

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024