A333322 Decimal expansion of (3/8) * sqrt(3).

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

Offset: 0

Author

Kritsada Moomuang, Mar 15 2020

Keywords

• cons
• nonn

Comments

This is the area of the regular hexagon of diameter 1.

From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)

For any triangle ABC, where (A,B,C) are the angles:

sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],

cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],

and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:

(ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).

The equalities are obtained only when triangle ABC is equilateral. (End)

Examples

			0.649519052838328985...
		

References

• O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.
• D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

Links

• Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

Crossrefs

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

Programs

Mathematica

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

PARI

sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

Formula

Equals A104954/2 or A104956/4.